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Sequence Theory | Math fields | Oct 3, 2021 | Alan Jo | Alan Jo | Oct 17, 2022 | [Set Theory](https://texonom.com/set-theory-a74b3919af5e4e6e885862f6f76103f9) |
๋์ด ์์๋ฅผ ์๊ฐํด์ผ ํ๊ณ ์ค๋ณต์ด ํ์ฉ๋๋ค๋ ์ ์์ ์งํฉ๊ณผ ๊ตฌ๋ถ๋๋ค
### Sequence Theory Notion
|Title|
|:-:|
|[Sequence](https://texonom.com/sequence-638838659a104ce3acc741ff806adf79)|
|[Series](https://texonom.com/series-da206bccff4241e7afb1c73a3aac306d)|
|[Recurrence relation](https://texonom.com/recurrence-relation-1c0b90005c30437fa61c5e049782e4be)|
| 708a6d7cc6e142f4b77c20c4f5179840 |
Topology | Math fields | Apr 11, 2023 | Alan Jo | Alan Jo | Jun 4, 2023 |
### Topology Notion
|Title|
|:-:|
|[Manifold](https://texonom.com/manifold-c6aa008ada9148309693c43c5033ad18)|
|[Closure](https://texonom.com/closure-f8ea031aff2a42e28ce310accdd3f38a)|
|[Compactness](https://texonom.com/compactness-4548634a95f645d2bf9f2027de247d5d)|
|[Heine Borel Theorem](https://texonom.com/heine-borel-theorem-9364af6db0584563850234356131360e)|
|[Poincare Conjecture](https://texonom.com/poincare-conjecture-ed72bbeef3774339adc4c3dfc7e83282)|
|[Euclidean Topology](https://texonom.com/euclidean-topology-d6b8e43b16264d8aaa37f576522e7063)|
> [Learn Topology in 5 minutes (joke video)](https://www.youtube.com/watch?v=aewo8otGAAQ&t=150s)
> [Science Is Fun on Twitter](https://twitter.com/sciencefunn/status/1641792459633205249)
| 3372655152c9472196ae94bbe7e15d6a |
|
**Algebraic Structure** | Algebra Areas | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | [Category Theory](https://texonom.com/category-theory-1b6594b7d4ce4c128ec2f9aed95d9f3c) |
### **Algebraic Structures**
|Title|
|:-:|
|[Hybrid structure](https://texonom.com/hybrid-structure-424a11fa9327407ba3ed9903ae38b504)|
|[One set with operations](https://texonom.com/one-set-with-operations-27c6050977164d3b9c27a6229596509b)|
|[Two sets with operations](https://texonom.com/two-sets-with-operations-0bb7f0b53f034a77b775132670be6fcc)|
### **Algebraic Structures Notion**
|Title|
|:-:|
|[Universal algebra](https://texonom.com/universal-algebra-7bd78848cc084965842680d528cd6934)|
|[Abstract Algebra](https://texonom.com/abstract-algebra-8337c2d030e84869a1e8b3a80c7e1007)|
> [Algebraic structure](https://en.wikipedia.org/wiki/Algebraic_structure)
| 4fb610f539064ec18d3edf2b310b7759 |
Lie Algebra | Algebra Areas | May 13, 2023 | Alan Jo | Alan Jo | May 13, 2023 |
### Lie Algebra Notion
|Title|
|:-:|
|[Lie group](https://texonom.com/lie-group-023939b50ef94e1a90d5475ca32ffe9a)|
| 77e6ab22461948778fff08818171cdae |
|
Linear Algebra | null | null | null | null | null | null |
### Linear Algebra Notion
|Title|
|:-:|
|[Vector](https://texonom.com/vector-9ed4d1c8519c496dbfadb1f6b3522255)|
|[Matrix](https://texonom.com/matrix-cbbb0614f0214e9ab9fd092a3b818132)|
|[Dimension](https://texonom.com/dimension-2e25a3a17e2d40eba201a337505a001d)|
|[Tensor](https://texonom.com/tensor-3fb2021969c44483945bacdae2904688)|
|[Kernel](https://texonom.com/kernel-13a25bedcee94977b02356d2320c490c)|
### Linear Algebra Usages
|Title|
|:-:|
|[BLAS](https://texonom.com/blas-17d8d7491b2147829b88426574733651)|
|[Vectorious](https://texonom.com/vectorious-fd5a587523a9423089cf74ae0899b8ce)|
### English
> [Essence of linear algebra - YouTube](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)
### Korean
> [ํ๋ ฌ๊ณผ ์ ํ๋ณํ](https://angeloyeo.github.io/2019/07/15/Matrix_as_Linear_Transformation.html)
> ['Fundamentals/Linear Algebra' ์นดํ
๊ณ ๋ฆฌ์ ๊ธ ๋ชฉ๋ก](https://twlab.tistory.com/category/Fundamentals/Linear%20Algebra)
| a5879da463c442d89c51891d124f97ac |
**Universal algebra** | Algebra Areas | May 13, 2023 | Alan Jo | Alan Jo | May 13, 2023 | 8f94ab573ea2438b84deac618daafeda |
||
**Hybrid structure** | Algebraic Structures | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
### **Hybrid structures**
|Title|
|:-:|
| 424a11fa9327407ba3ed9903ae38b504 |
|
One set with operations | Algebraic Structures | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
### One set Algebraic Structure
|Title|
|:-:|
|[Set](https://texonom.com/set-108ac7e311584c39be1e11102cada616)|
|[Group theory](https://texonom.com/group-theory-8aaba49fbf634cffb02439c1f0f5c3b0)|
|[Ring theory](https://texonom.com/ring-theory-0c4a54c6d758443eba9a112b6dfa2e05)|
|[Lattice Theory](https://texonom.com/lattice-theory-b8892bfb58be4088b07f871c8ce75884)|
| 27c6050977164d3b9c27a6229596509b |
|
****Two sets with operations**** | null | null | null | null | null | null |
### Two set Algebraic Structures
|Title|
|:-:|
|[Module Theory](https://texonom.com/module-theory-1301a52f9e594bb9902de69c194e4505)|
|[Algebra over a field](https://texonom.com/algebra-over-a-field-c2685b5b35184fefa37039df0fa2ec2d)|
| 0bb7f0b53f034a77b775132670be6fcc |
**Group theory** | One set Algebraic Structure | May 13, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 |
## Group-like Structure
**one**ย binary operation
### **Group theory Notion**
|Title|
|:-:|
|[Abelian Group](https://texonom.com/abelian-group-02931d2de53c4cd69eced1eab6329860)|
|[Nilpotent group](https://texonom.com/nilpotent-group-af68f13e50f94eca85b000782b60371f)|
|[Dihedral Group](https://texonom.com/dihedral-group-309c615786f84b4e805b4413429534d1)|
|[Abelian Group](https://texonom.com/abelian-group-4878c844bde74e53914d3d06eb98db3d)|
|[Lie Group](https://texonom.com/lie-group-89e94fbc92af4bbe925034210f029e22)|
|[Fundamental group](https://texonom.com/fundamental-group-67245af792954c3e8e32e41ca1dcf49a)|
|[Topological group](https://texonom.com/topological-group-637e9ea8b7c448fe925b699488cf00e4)|
> [Group theory](https://en.wikipedia.org/wiki/Group_theory)
> [An Introduction To Group Theory](https://www.youtube.com/watch?v=zkADn-9wEgc)
> [Euler's formula with introductory group theory](https://youtu.be/mvmuCPvRoWQ)
| 8aaba49fbf634cffb02439c1f0f5c3b0 |
|
Lattice Theory | One set Algebraic Structure | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
### Lattice Theory Notion
|Title|
|:-:|
|[Complete Lattice](https://texonom.com/complete-lattice-17bf5dfa979646ec85ae4fd6508c0aa2)|
|[Bounded Lattice](https://texonom.com/bounded-lattice-f52c77f86652473bbd02762c0211ff82)|
|[Distributive Lattice](https://texonom.com/distributive-lattice-6490ba5f4efa458bb8982b177d56af02)|
| b8892bfb58be4088b07f871c8ce75884 |
|
**Ring theory** | One set Algebraic Structure | May 13, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
### **Ringoids, Ring-like structures**
two binary operations
### **Ring theory Notion**
|Title|
|:-:|
|[Field](https://texonom.com/field-a539c71297824a56a0a7139c871aaa08)|
|[Division ring](https://texonom.com/division-ring-ab03adc9465d4fbeb6dd568175d6489d)|
|[Commutative Ring](https://texonom.com/commutative-ring-a0cf1c6c1b9049bea92cb76b53645ce9)|
|[Nilpotent](https://texonom.com/nilpotent-fda8c11879cb4115a1f1ab85e77b3216)|
|[Unipotent](https://texonom.com/unipotent-a85dbc60c7cb4f8da7e289ea63a728f7)|
|[Nil ideal](https://texonom.com/nil-ideal-173d77e30c4445569ffd1edb37134a63)|
|[Scheme](https://texonom.com/scheme-08622f9840264c049ecd66d8d224d613)|
> [Ring theory](https://en.wikipedia.org/wiki/Ring_theory)
> [Ring Theory 1: Ring Definition and Examples](https://www.youtube.com/watch?v=-ljA4ZsX5Es)
| 0c4a54c6d758443eba9a112b6dfa2e05 |
|
Set | null | null | null | null | null | null |
### Simple Structure, no binary operation
| 108ac7e311584c39be1e11102cada616 |
Abelian Group | Group theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
a group whose binary operation is [Commutative Property](https://texonom.com/commutative-property-216366d94b2d43fb87d11e87cc3a233a)
| 02931d2de53c4cd69eced1eab6329860 |
|
Abelian Group | Group theory Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 |
### commutative group
[Commutative Property](https://texonom.com/commutative-property-216366d94b2d43fb87d11e87cc3a233a) ์ฑ๋ฆฝ๊ตฐ
์ ์ํ ์์ ๊ฐํ๊ตฐ
| 4878c844bde74e53914d3d06eb98db3d |
|
Dihedral Group | Group theory Notion | Jun 12, 2023 | Alan Jo | Alan Jo | Jun 12, 2023 | 309c615786f84b4e805b4413429534d1 |
||
Fundamental group | Group theory Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 | [Henri Poincare](https://texonom.com/henri-poincare-1382da083d864065972f27c140c4aa64) |
### Poincare Group
| 67245af792954c3e8e32e41ca1dcf49a |
Lie Group | Group theory Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 |
### Lie Group Types
|Title|
|:-:|
|[Ree group](https://texonom.com/ree-group-ec739cf283924fc5af124908dc7f3cbe)|
| 89e94fbc92af4bbe925034210f029e22 |
|
**Nilpotent group** | Group theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | af68f13e50f94eca85b000782b60371f |
||
Topological group | Group theory Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 |
### **์์๊ตฐ**
| 637e9ea8b7c448fe925b699488cf00e4 |
|
**Ree group** | Lie Group Types | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 |
> [Ree group](https://en.wikipedia.org/wiki/Ree_group)
| ec739cf283924fc5af124908dc7f3cbe |
|
Bounded Lattice | Lattice Theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | f52c77f86652473bbd02762c0211ff82 |
||
Complete Lattice | Lattice Theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | 17bf5dfa979646ec85ae4fd6508c0aa2 |
||
Distributive Lattice | Lattice Theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | 6490ba5f4efa458bb8982b177d56af02 |
||
Commutative Ring | null | null | null | null | null | null |
### Commutative Rings
|Title|
|:-:|
|[Dual number](https://texonom.com/dual-number-486c2770b7e5428594cf455e3de288a5)|
| a0cf1c6c1b9049bea92cb76b53645ce9 |
Division ring | Ring theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | ab03adc9465d4fbeb6dd568175d6489d |
||
Field | null | null | null | null | null | null |
free [Elementary arithmetic](https://texonom.com/elementary-arithmetic-57a9f00b3f30433299816f32b4d49fbd)
every field is [Commutative Ring](https://texonom.com/commutative-ring-a0cf1c6c1b9049bea92cb76b53645ce9)
### Field Algebra Notion
|Title|
|:-:|
|[Algebraically Closed Field](https://texonom.com/algebraically-closed-field-a7ae2a1584f04bcfab20c148c98113cd)|
|[Algebraic closure](https://texonom.com/algebraic-closure-896e8f79e2654e8fa56e0381916e76c4)|
### Fields
|Title|
|:-:|
|[Vector Field](https://texonom.com/vector-field-ff7ea488edb74510a51e844eee7a222f)|
|[Scalar Field](https://texonom.com/scalar-field-a0e05a5b310c4a6283bc214ec02383e0)|
| a539c71297824a56a0a7139c871aaa08 |
**Nil ideal** | Ring theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | 173d77e30c4445569ffd1edb37134a63 |
||
Nilpotent | Ring theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
๊ฑฐ๋ญ์ ๊ณฑํ์ฌ 0์ด ๋๋ ์์
> [Nilpotent](https://en.wikipedia.org/wiki/Nilpotent)
| fda8c11879cb4115a1f1ab85e77b3216 |
|
Scheme | Ring theory Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 | [Alexandre Grothendieck](https://texonom.com/alexandre-grothendieck-e42d368dff3d42afa53cd2b1132bdedf) [Commutative Ring](https://texonom.com/commutative-ring-a0cf1c6c1b9049bea92cb76b53645ce9) |
### Ringsโ [Zariski gluing](https://texonom.com/zariski-gluing-690c524648eb40b08360d0c87bceee7d)
### Scheme Notion
|Title|
|:-:|
|[Sheaf](https://texonom.com/sheaf-c1aa0624425d40ab86351c509a86b389)|
> [Scheme (mathematics)](https://en.wikipedia.org/wiki/Scheme_(mathematics))
| 08622f9840264c049ecd66d8d224d613 |
**Unipotent** | Ring theory Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | a85dbc60c7cb4f8da7e289ea63a728f7 |
||
Dual number | Commutative Rings | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
[Real Number](https://texonom.com/real-number-2c5c6fff21ab429da38ba685b40bacf3) and $\epsilon^2 = 0 (\epsilon \ne 0)$
> [Dual number](https://en.wikipedia.org/wiki/Dual_number)
| 486c2770b7e5428594cf455e3de288a5 |
|
**Algebraic closure** | Field Algebra Notion | Jun 7, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 | 896e8f79e2654e8fa56e0381916e76c4 |
||
Algebraically Closed Field | Field Algebra Notion | Jun 7, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 | a7ae2a1584f04bcfab20c148c98113cd |
||
Scalar Field | Fields | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 |
**์ ํด๋ฆฌ๋ ๊ณต๊ฐ์ ๊ฐ ์ ์ ์ค์นผ๋ผ๋ฅผ ๋์์ํจ ๊ฒ**
| a0e05a5b310c4a6283bc214ec02383e0 |
|
Vector Field | Fields | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 |
### Vector Fields
|Title|
|:-:|
|[Conservative vector field](https://texonom.com/conservative-vector-field-83aa7666a0134c1189bbc69aaa79c8a0)|
| ff7ea488edb74510a51e844eee7a222f |
|
**Conservative vector field** | Vector Fields | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 |
### irrotational
| 83aa7666a0134c1189bbc69aaa79c8a0 |
|
Sheaf | Scheme Notion | Jul 12, 2023 | Alan Jo | Alan Jo | Jul 12, 2023 | [Sheaf cohomology](https://texonom.com/sheaf-cohomology-34a60e81bccb4761bcaac43c86e75d72) | c1aa0624425d40ab86351c509a86b389 |
|
**Algebra over a field** | Two set Algebraic Structures | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | c2685b5b35184fefa37039df0fa2ec2d |
||
Module Theory | Two set Algebraic Structures | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 |
an abelian groupย *M*ย and a ringย *R*ย acting as operators onย *M*
### Module Theory Notion
|Title|
|:-:|
| 1301a52f9e594bb9902de69c194e4505 |
|
Abstract Algebra | null | null | null | null | null | null |
### Abstract Algebra Notion
|Title|
|:-:|
> [Abstract algebra](https://en.wikipedia.org/wiki/Abstract_algebra)
| 8337c2d030e84869a1e8b3a80c7e1007 |
**Universal algebra** | Algebraic Structures Notion | Jun 6, 2023 | Alan Jo | Alan Jo | Jun 6, 2023 | [Category Theory](https://texonom.com/category-theory-1b6594b7d4ce4c128ec2f9aed95d9f3c) |
### **Universal algebra Notion**
|Title|
|:-:|
> [Universal algebra](https://en.wikipedia.org/wiki/Universal_algebra)
| 7bd78848cc084965842680d528cd6934 |
**Lie group** | Lie Algebra Notion | May 13, 2023 | Alan Jo | Alan Jo | May 13, 2023 | 023939b50ef94e1a90d5475ca32ffe9a |
||
Dimension | null | null | null | null | null | null |
๋์ด๋ก ์ ๋ฐ๋ฅด๋ฉด 10์ฐจ์ ํน์ ๊ทธ ์ด์
### Dimension Notion
|Title|
|:-:|
|[Curse of dimensionality](https://texonom.com/curse-of-dimensionality-343455a61aa14779b6edf4c93464fac3)|
|[Tesseract](https://texonom.com/tesseract-82c23327fd724b059982d4cd4e060971)|
### Dimension Usages
|Title|
|:-:|
|[Observed dimensionality](https://texonom.com/observed-dimensionality-33fd6337a4604a0e9a2d6b04c346d16e)|
> [n์ฐจ์ ์ธ๊ณ์์ ์ผ์ด๋๋ ๋ฏฟ์ ์ ์๋ ์ ๊ธฐํ ํ์!!](https://youtube.com/watch?v=EXHR2-hECRM&feature=shares)
| 2e25a3a17e2d40eba201a337505a001d |
Kernel | Linear Algebra Notion | Apr 17, 2023 | Alan Jo | Alan Jo | Jun 5, 2023 |
## **nullspace**
Linear subspace of the domain of the map which is mapped into the zero vector
v๊ฐ u์ ๋ํ Orthogonal ๊ด๊ณ๋ฅผ ๊ฐ๋๋ค๋ฉด, v๋ u์ Kernel์ด๋ผ๊ณ ํฉ๋๋ค.
$$Ker(u) = {v โ V | <u,v> = 0}$$
> [Kernel (linear algebra)](https://en.wikipedia.org/wiki/Kernel_(linear_algebra))
| 13a25bedcee94977b02356d2320c490c |
|
***Matrix*** | Linear Algebra Notion | Jan 20, 2022 | Alan Jo | Alan Jo | Jul 18, 2023 | [Vector](https://texonom.com/vector-9ed4d1c8519c496dbfadb1f6b3522255) |
## Matrix is a Linear transformation or Information
Usually small character is vector and capital character is Matrix
์ ํ ๋ณํ์ ๊ธฐํํ์ ์ผ๋ก ์๊ฐํ๋ฉด ์ผ์ข
์ ๊ณต๊ฐ ๋ณํ
### ***Matrix Notion***
|Title|
|:-:|
|[Matrix Operation](https://texonom.com/matrix-operation-49662a8ed4724c89b3f98e38b3df7069)|
|[Inverse Matrix](https://texonom.com/inverse-matrix-737012fc4e50403696dfa9bb16e20b7f)|
|[Eigenvector](https://texonom.com/eigenvector-d2424f7eaea4443d81105606ebf4e975)|
|[Square Matrix](https://texonom.com/square-matrix-21e528744f2e4d81910de180aa0157ce)|
|[Transpose of Matrix](https://texonom.com/transpose-of-matrix-639bc5b5563c4c54b3e9ef0f37342bee)|
|[Matrix Basis](https://texonom.com/matrix-basis-38e395efb4674efbaf42f5b01333aab3)|
|[Define Matrix](https://texonom.com/define-matrix-30084846a14044df871882b9ba2d1882)|
|[Orthogonal Matrix](https://texonom.com/orthogonal-matrix-5e743532b7ac4364b6c1737e2da53798)|
|[Matrix Decomposition](https://texonom.com/matrix-decomposition-dd4db9ba7abb480eb609957095c4e7d7)|
|[Triangular Matrix](https://texonom.com/triangular-matrix-6d442a1dc56346ccb417b1ee4d23ef52)|
|[Positive Definite Matrix](https://texonom.com/positive-definite-matrix-4b9b4c65a6ec4cd5b86e4c3216f3ef94)|
### ***Matrix Usages***
|Title|
|:-:|
|[Matrix Derivative](https://texonom.com/matrix-derivative-67e3bac465684331aa5205893b542749)|
|[Direction Cosine Matrix](https://texonom.com/direction-cosine-matrix-0622b233d72a4ad68a923fe7f8a806be)|
|[Matrix Computation](https://texonom.com/matrix-computation-af84cd99a7d047caa70d3517e26f73e3)|
|[Covariance Matrix](https://texonom.com/covariance-matrix-9067b8dc2bbc43ce915c4107eaf84bb5)|
|[Matrix Padding](https://texonom.com/matrix-padding-0f44aaaee9d54bb3a6960130541ac90a)|
|[Jacobian Matrix](https://texonom.com/jacobian-matrix-3d36041b4bf64dcd96bd06ff680fe810)|
|[Correlation Matrix](https://texonom.com/correlation-matrix-fefce53c8a2c4d91b111167e7d0e4094)|
| cbbb0614f0214e9ab9fd092a3b818132 |
Tensor | Linear Algebra Notion | Mar 14, 2023 | Alan Jo | Alan Jo | Sep 16, 2023 |
[Vector](https://texonom.com/vector-9ed4d1c8519c496dbfadb1f6b3522255), [Matrix](https://texonom.com/matrix-cbbb0614f0214e9ab9fd092a3b818132) ๊ณ์ฐ์
usually vector means 1d and matrix means 2d
tensor is term for any dimensional data
[Kroneckerย Delta](https://texonom.com/kroneckerdelta-5e1b070c308841eb831b512ef64664f3)
> [How a Computer Broke a 50-Year Math Record](https://www.youtube.com/watch?v=fDAPJ7rvcUw)
| 3fb2021969c44483945bacdae2904688 |
|
Vector | Linear Algebra Notion | Nov 5, 2019 | Alan Jo | Seong-lae Cho | Jul 10, 2023 | [Matrix](https://texonom.com/matrix-cbbb0614f0214e9ab9fd092a3b818132) [Vector Calculus](https://texonom.com/vector-calculus-77976c6127ad49c9a6611e16a40f337f) |
## magnitude + direction
### Vector Notion
|Title|
|:-:|
|[Vector Operation](https://texonom.com/vector-operation-61d2aa5fe6594af7a30b45309946b2d7)|
|[Linear Independent](https://texonom.com/linear-independent-da490558cce24664be62d499d40b96a2)|
|[Unit Vector](https://texonom.com/unit-vector-9d0e1e60888c4f338745830b40584981)|
|[Vector Space](https://texonom.com/vector-space-7681a959944242619d201953076294f2)|
|[Isotropic](https://texonom.com/isotropic-13d450b4d2d24ef18af47546f578c3e7)|
|[Anisotropic](https://texonom.com/anisotropic-56427ad6cfc84d46a0b46606b25bb7bd)|
### Vector Usages
|Title|
|:-:|
|[Vector Equation](https://texonom.com/vector-equation-122ad1c33c514ad3b2c9cee4aac00521)|
|[Vector Similarity](https://texonom.com/vector-similarity-846675a408e74235acc836302110ab07)|
| 9ed4d1c8519c496dbfadb1f6b3522255 |
Curse of dimensionality | null | null | null | null | null | null |
์ ์ฐจ์ ๊ณต๊ฐ์์ ์ป์ ์ง๊ด์ด ๊ณ ์ฐจ์ ๊ณต๊ฐ์์๋ ํต์ฉ๋ ๊ฒ์ด๋ผ๋ ๊ฒ์ ๋ฏฟ์ผ๋ฉด ์๋๋ค
์ฐจ์์ด ์ฆ๊ฐํ๋ฉด ๊ณต๊ฐ์ ๋ถํผ๊ฐ ๋๋ฌด ๋นจ๋ฆฌ ์ฆ๊ฐํ์ฌ ์ฌ์ฉ ๊ฐ๋ฅํ ๋ฐ์ดํฐ๊ฐ ํฌ์ํด์ง๋ค๋ ๊ฒ์ด๋ค. ์ ๋ขฐํ ์ ์๋ ๊ฒฐ๊ณผ๋ฅผ ์ป๊ธฐ ์ํด ํ์ํ ๋ฐ์ดํฐ์ ์์ด ์ฐจ์์ ๋ฐ๋ผ ๊ธฐํ๊ธ์์ ์ผ๋ก ์ฆ๊ฐํ๋ ๊ฒฝ์ฐ
> [n์ฐจ์ ์ธ๊ณ์์ ์ผ์ด๋๋ ๋ฏฟ์ ์ ์๋ ์ ๊ธฐํ ํ์!!](https://www.youtube.com/watch?v=EXHR2-hECRM&t=630s)
> [4. The Curse of Dimensionality](http://norman3.github.io/prml/docs/chapter01/4)
| 343455a61aa14779b6edf4c93464fac3 |
Tesseract | Dimension Notion | Mar 28, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 |
์์ ์์ฌ๋ก ๋ง์ด ์ฐ์
์ด์
๋ฐฉ์ฒด
3์ฐจ์ ์ ๊ฐ๋๋ก ๋ง์ด ์ฐ
| 82c23327fd724b059982d4cd4e060971 |
|
Observed dimensionality | null | null | null | null | null | null |
๋ฐ์ดํฐ์
์์ ์ค์ ๋ก ๊ด์ฐฐ๋๋ ์ฐจ์์ ์
๋ฐ์ดํฐ์
์ ๋ณดํต ๋ชจ๋ ๊ฐ๋ฅํ ๊ฒฝ์ฐ๋ฅผ ํฌํจํ์ง ์์ผ๋ฏ๋ก (์๋ฏธ์๋ ๋ฐ์ดํฐ๋ ๋ฏธํฌํจ). ๋ฐ๋ผ์ ์ด๋ฌํ ๊ฒฝ์ฐ์๋ ๋ฐ์ดํฐ์
์ ์ฐจ์์ด ์ค์ ๋ก๋ ๋ฎ์ถ์ ์๋ค
| 33fd6337a4604a0e9a2d6b04c346d16e |
Define Matrix | Matrix Notion | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 |
### Define Matrix Notion
|Title|
|:-:|
|[Positive Definite](https://texonom.com/positive-definite-ecaeb51005c0411a8f515f5eb8612c6d)|
|[Positive semi-definite](https://texonom.com/positive-semi-definite-402fa18c1ec74c7a80caba4b1c95cc3d)|
| 30084846a14044df871882b9ba2d1882 |
|
Eigenvector | null | null | null | null | null | null |
### ๊ณ ์ ๋ฒกํฐ
**ํ๋ ฌ(์ ํ๋ณํ)์ด ๋ฒกํฐ์ ์์ฉํ๋ principal axis**
$$Me = \lambda e$$
1. Compute the determinant of $M - \lambda I$ which returns a polynomial
2. Find the roots of polynomial $det(M - \lambda I) = 0$ which returns eigenvalues
3. For each eigenvalue, solve $(M - \lambda I)e = 0$ which returns eigenvectors
box๋ค์ ํฌ์ธํธ๋ค์ ๋ํด eigen vector๋ฅผ ๊ตฌํ๋ค์, eigen value๊ฐ ์์ 2๊ฐ์ ๋ฒกํฐ๊ฐ ํ๋ฉด์ ํํํ ๋ฒกํฐ๋ผ ๊ฐ์
๊ณ ์ ๋ฒกํฐ์ ๋ฐ์ดํฐ๋ฅผ ์ ์ฌ์ํ์ฌ ์ป์ data์ variance๊ฐ ์ต๋๊ฐ ๋๋ค
### Eigenvalue Notion
|Title|
|:-:|
|[Eigenvalue Equation](https://texonom.com/eigenvalue-equation-db381bfd94bd4e5990893ba1579a59fc)|
|[Eigenvalue](https://texonom.com/eigenvalue-4dff6b022dd94d19b3791eef258bf9c5)|
> [[์ ํ๋์ํ #3] ๊ณ ์ ๊ฐ๊ณผ ๊ณ ์ ๋ฒกํฐ (eigenvalue & eigenvector)](https://darkpgmr.tistory.com/105)
| d2424f7eaea4443d81105606ebf4e975 |
Inverse Matrix | Matrix Notion | Apr 17, 2023 | Alan Jo | Alan Jo | Sep 13, 2023 | [Square Matrix](https://texonom.com/square-matrix-21e528744f2e4d81910de180aa0157ce) |
### Inverse Matrix Notion
|Title|
|:-:|
|[Invertible Matrix Theorem](https://texonom.com/invertible-matrix-theorem-4eb11f1ca7c54a9e8bd081b28aeb5b33)|
| 737012fc4e50403696dfa9bb16e20b7f |
Matrix Basis | null | null | null | null | null | null |
[Matrix Rank](https://texonom.com/matrix-rank-1bc59c1dd997400cb5072de4ed588495)
### Template Gallery
|Title|
|:-:|
|[Template Page](https://texonom.com/template-page-6ab9aa76bfe0497b8861fb439560e38f)|
| 38e395efb4674efbaf42f5b01333aab3 |
Matrix Decomposition | Matrix Notion | Apr 13, 2023 | Alan Jo | Alan Jo | Jul 18, 2023 |
### Matrix Decompositions
|Title|
|:-:|
|[SVD](https://texonom.com/svd-c7cdeb1b6b4747c8825b7e23c325d1ab)|
|[QR Decomposition](https://texonom.com/qr-decomposition-b2e297ba8eda4bbaa7eb8761a2ab7cff)|
|[Cholesky decomposition](https://texonom.com/cholesky-decomposition-0e9242858d9d4c97a1d811527b8a17c8)|
| dd4db9ba7abb480eb609957095c4e7d7 |
|
Matrix Operation | Matrix Notion | Sep 26, 2022 | Alan Jo | Alan Jo | Apr 17, 2023 | [Transpose of Matrix](https://texonom.com/transpose-of-matrix-639bc5b5563c4c54b3e9ef0f37342bee) |
### Matrix Operaters
|Title|
|:-:|
|[Matrix Multiplication](https://texonom.com/matrix-multiplication-b2a5fb3a1d354f7c990b2ea3bcfcb751)|
|[Matrix Trace](https://texonom.com/matrix-trace-0b7fd1b5735549f782a13909912b02a9)|
|[Matrix Square](https://texonom.com/matrix-square-414ae2d42d0e45df99d95847c99c0116)|
> [Untitled](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf)
| 49662a8ed4724c89b3f98e38b3df7069 |
Orthogonal Matrix | null | null | null | null | null | null |
## Orthonormal Matrix
[Orthonormal Vectors](https://texonom.com/orthonormal-vectors-6c05c455048d4d729c7a86ad3802b339) ๋ก ๊ตฌ์ฑ. [Identity matrix](https://texonom.com/identity-matrix-30db8dd3fc3b4e7da2d12cbcfc68a25b)๊ฐ ๋ํ์
$$A^T = A^{-1}$$
### Orthogonal Matrix Usages
|Title|
|:-:|
|[Gram-Schmidt Process](https://texonom.com/gram-schmidt-process-60e50ee9d9f042b986cafaf3e0e960b3)|
> [[Linear Algebra] Lecture 17-(1) ์ง๊ตํ๋ ฌ(Orthogonal Matrices)๊ณผ ๊ทธ๋ ์๋ฏธํธ ๊ณผ์ (Gram-Schmidt Process)](https://twlab.tistory.com/37)
| 5e743532b7ac4364b6c1737e2da53798 |
Positive Definite Matrix | Matrix Notion | Jan 23, 2022 | Alan Jo | Alan Jo | Apr 17, 2023 | [Hessian Matrix](https://texonom.com/hessian-matrix-e1ebbf5284284a1793233973648ef0b6) |
์๋ฒกํฐ๊ฐ ์๋ ์์์ ์ด๋ฒกํฐย x์ ๋์นญ ํ๋ ฌย A์ ๋ํด ๋ค์์ด ์ฑ๋ฆฝํ๋ A
$$x^TAx > 0$$
์์ ์ ๋ถํธ ์ฌ๋ถ๊ฐ ๊ฐ์ฅ ์ค์ํ๊ฒ ์ฌ์ฉ๋๋ ์์ฉ ๋ถ์ผ ์ค ํ๋๋ ์ต์ ํ ๋ถ์ผ
๋ ๋ฒ ๋ฏธ๋ถ๊ฐ๋ฅํ ์คํจ์ย f ์ ๋ํด Hessian ํ๋ ฌ์ด ์์ ์ ๋ถํธ ํ๋ ฌ์ด๋ผ๋ฉด ์ด ํจ์์ ์ ์ฒด์ ์ธ ํํ๋ ์๋๋ก ๋ณผ๋กํ๊ณ , ํ์ฐ์ ์ผ๋ก ๊ทน์๊ฐ์ ๊ฐ๋๋ค
๋ง์ฝ ์ฌ๊ธฐ์ gradient๋ฅผ ๊ตฌํด์ 0์ธ ์ง์ ์ ์ฐพ๋๋ค๋ฉด ๊ทธ ํฌ์ธํธ๊ฐ ๋ฐ๋ก ๊ทน์๊ฐ
> [์์ ์ ๋ถํธ (positive definite) ํ๋ ฌ](https://www.youtube.com/watch?v=fuxJ-VH8pWk)
> [์์ ์ ๋ถํธ ํ๋ ฌ (positive definite matrix)](https://angeloyeo.github.io/2021/12/20/positive_definite.html)
| 4b9b4c65a6ec4cd5b86e4c3216f3ef94 |
Square Matrix | Matrix Notion | Apr 6, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 |
### Square Matrixes
|Title|
|:-:|
|[Symmetric Matrix](https://texonom.com/symmetric-matrix-8ab18b2c502545e08445c0110a090131)|
|[Identity matrix](https://texonom.com/identity-matrix-30db8dd3fc3b4e7da2d12cbcfc68a25b)|
|[Diagonal Matrix](https://texonom.com/diagonal-matrix-622ba1ae252f4e3fbe1328e155430a64)|
|[Toeplitz Matrix](https://texonom.com/toeplitz-matrix-7d66a501e56c4cc8b407dc9a586724c4)|
|[Circulant matrix](https://texonom.com/circulant-matrix-0fcb400b1bd84d37876a8e1340a25176)|
### Square Matrix Usages
|Title|
|:-:|
|[Determinant](https://texonom.com/determinant-92e4c12610184c8e91a869761d7a5c8c)|
| 21e528744f2e4d81910de180aa0157ce |
|
Transpose of Matrix | Matrix Notion | Oct 28, 2022 | Alan Jo | Alan Jo | Apr 17, 2023 |
Matrix composed with 3 axis unit vectors x, y, zโs inverse matrix is an Identity matrix
$$\begin{pmatrix}\vec{x} & \vec{y} & \vec{z}\end{pmatrix}\begin{pmatrix}\vec{x} \\ \vec{y} \\ \vec{z}\end{pmatrix}= I$$
### But
I felt weird and realized it is different from normal matrix multiplication
- why vector is in matrix?
- why use dot product instead of number multiplication? Is it valid?
$$(AB)^T = B^TA^T$$
$$(a^TBc)^T = c^TB^Ta$$
$$a^T b = b^Ta$$
๊ณฑํ ๋ Transpose ์๋ค์ ์ฉํ ์ง ์ ํํ ๋ ์ผ๋จ ํ๋ ฌํฌ๊ธฐ ๋จผ์ ๊ณ ๋ คํ๊ณ ๋ํ๊ณ ์์ ๋นผ๋ ๋ค๋ฅธ ํญ์
๊ทธ๋ค์ ํฌ๊ธฐ๊ฐ ์์ ๊ฒฐ๊ณผ ํ๋ ฌ์ ์ ํํ์ฌ ๊ณ์ฐํ๋ ๊ฒ์ด ์ข๋ค
๋ณดํต ์์ ๊ฒ transpose
### Diverse Attributes
> [Transpose](https://en.wikipedia.org/wiki/Transpose)
> [Untitled](http://www.gatsby.ucl.ac.uk/teaching/courses/sntn/sntn-2017/resources/Matrix_derivatives_cribsheet.pdf)
| 639bc5b5563c4c54b3e9ef0f37342bee |
|
Triangular Matrix | Matrix Notion | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 |
์ผ๊ฐํ ๋ชจ์ ๋ถ๋ถ ๋นผ๊ณ ๋๋จธ์ง๋ 0
### Triangular Matrixes
|Title|
|:-:|
|[Upper Triangular Matrix](https://texonom.com/upper-triangular-matrix-9306a963fe274b20b34438719d709f2f)|
|[Lower Triangular Matrix](https://texonom.com/lower-triangular-matrix-68d11496d8614893ab3f3fcedf150a23)|
> [Triangular matrix](https://en.wikipedia.org/wiki/Triangular_matrix)
| 6d442a1dc56346ccb417b1ee4d23ef52 |
|
Positive Definite | Define Matrix Notion | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 | ecaeb51005c0411a8f515f5eb8612c6d |
||
Positive semi-definite | Define Matrix Notion | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 |
### ์์ ์ค์ ๋ถํธ
[Eigenvalue](https://texonom.com/eigenvalue-4dff6b022dd94d19b3791eef258bf9c5) is always bigger than 0
### Positive semi-definites
|Title|
|:-:|
|[Symmetric positive semi-definite](https://texonom.com/symmetric-positive-semi-definite-16ce5c2766e94f3d803ea6ddf2742660)|
| 402fa18c1ec74c7a80caba4b1c95cc3d |
|
Symmetric positive semi-definite | Positive semi-definites | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 |
์กฐ๊ฑด๋ง์กฑํ๋ฉด ์ฌ๊ธฐ์ ๊ธฐ ์ฐ์ธ๋ค
| 16ce5c2766e94f3d803ea6ddf2742660 |
|
Eigenvalue | Eigenvalue Notion | Jan 23, 2022 | Alan Jo | Alan Jo | Jul 18, 2023 |
Each eigenvalue $\lambda$ is associated with a specific eigenvector $\vec{x}$ for A
$$A\vec{x} = \lambda\vec{x}$$
Eigenvalue show how much eigenvector change to the direction
eigenvalue๊ฐ ์์์๋ก ์ญํ๋ ฌ์ ํฌ๊ธฐ๋ ์ปค์ง (eigenvalue ์ญ์๊ด๊ณ)
๊ทธ๋ฌ๋ eigenvalue๊ฐ 0์ ๊ฐ๊น์ธ์๋ก ์ญํ๋ ฌ์ ํฌ๊ธฐ๋ ๋งค์ฐ ์ปค์ง๋ฏ๋ก, ๊ณ์ฐ์ ์ผ๋ก ๋ถ์์
> [๊ณ ์ณ๊ฐ๊ณผ ๊ณ ์ ๋ฒกํฐ](https://angeloyeo.github.io/2019/07/17/eigen_vector.html)
| 4dff6b022dd94d19b3791eef258bf9c5 |
|
Eigenvalue Equation | Eigenvalue Notion | Aug 31, 2021 | Alan Jo | Alan Jo | Mar 23, 2023 |
๊ณ ์ ์ญํ์ ๋ผ๊ทธ๋์ง ๋ฐฉ์ ์์ด๋, ์์์ญํ์ ์ฌ๋ขฐ๋ฉ๊ฑฐ ๋ฐฉ์ ์์ด ๋ฐ๋ก ๊ณ ์ ์น ๋ฐฉ์ ์
์์์ญํ์์ ๊ณ ์ ์น๋ ์ค์ ๋ก ์ธก์ ๊ฐ๋ฅํ ๊ฐ(๊ด์ธก๋๋ ๊ฐ)์ ์๋ฏธํ๊ณ ๊ณ ์ ๋ฒกํฐ๋ ์ํํจ์
์ธก์ ํ์ง ์์์๋์ ์ํ๋ ๊ณ ์ ๋ฒกํฐ์ ์ ํ๊ฒฐํฉ
๋ผ๊ทธ๋์ง ๋ฐฉ์ ์์ ์ผ๋ฐ์ธ๋ค์๊ฒ๋ ์ ๋ช
ํ ๋ดํผ ๋ฐฉ์ ์์ ๊ณ ์ ์น๋ฌธ์ ๋ก ๋ค๋ฃจ์ด์ ๋ฌผ์ฒด์ ์ด๋์ํ๋ฅผ ์ง๊ด์ ์ผ๋ก ์ดํด
์ฌ๊ธฐ์์ ๊ณ ์ ์น๋ ์ผ๋ฐํ์ขํ์ด๊ณ ์ค์ ์ ๋ฌผ์ฒด์ ์ด๋์ ์ด ์ผ๋ฐํ ์ขํ๋ค์ ์ ํ ๊ฒฐํฉ
๊ณ ์ ์น ๋ฐฉ์ ์์ ํ์ด ์ด ์ผ๋ฐํ ์ขํ๋ฅผ ์ฐพ์๋ด๊ณ ์ด ์ขํ์ ๋ฌผ๋ฆฌ์ ์๋ฏธ๋ฅผ ์ดํดํ๊ฒ ๋๋ฉด ๋ณต์กํ ์ด๋๋ ์์ฃผ ๋จ์ํ ์ด๋์ ๊ฒฐํฉ์ผ๋ก ์ดํด๊ฐ๋ฅ
> [[์ ํ๋์ํ #3] ๊ณ ์ ๊ฐ๊ณผ ๊ณ ์ ๋ฒกํฐ (eigenvalue & eigenvector)](https://darkpgmr.tistory.com/105)
| db381bfd94bd4e5990893ba1579a59fc |
|
Invertible Matrix Theorem | Inverse Matrix Notion | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 | 4eb11f1ca7c54a9e8bd081b28aeb5b33 |
||
Matrix Rank | Matrix Basis | null | null | null | null | null |
number of basis
[Matrix Rank Deficiency](https://texonom.com/matrix-rank-deficiency-0225d134d466420a85e38fd9eca45442)
| 1bc59c1dd997400cb5072de4ed588495 |
Matrix Rank Deficiency | Matrix Rank | null | null | null | null | null |
Theย **rank deficiency**ย of a matrix is the difference between the lesser of the number of rows and columns, and the rank
> [Rank (linear algebra)](https://en.wikipedia.org/wiki/Rank_(linear_algebra))
| 0225d134d466420a85e38fd9eca45442 |
Template Page | Template Gallery | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 6, 2023 | 6ab9aa76bfe0497b8861fb439560e38f |
||
****Cholesky decomposition**** | Matrix Decompositions | Aug 8, 2021 | Alan Jo | Alan Jo | Jun 25, 2023 | [Inverse Matrix](https://texonom.com/inverse-matrix-737012fc4e50403696dfa9bb16e20b7f) [Hermitian matrix](https://texonom.com/hermitian-matrix-15f653dc272b4438bcd9057ea60c1e7f) |
Linear System ํจ๊ณผ์ ์ผ๋ก
์ญํ๋ ฌ์ ๊ตฌํ๋ ค๊ณ ํ ๋, ์กฐ๊ธ ๋ ํจ์จ์ ์ผ๋ก ๊ตฌํ๊ธฐ ์ํด
> [Cholesky decomposition](https://en.wikipedia.org/wiki/Cholesky_decomposition)
> [Untitled](https://taeyoung96.github.io/slam/SLAM_02/)
| 0e9242858d9d4c97a1d811527b8a17c8 |
**QR Decomposition** | Matrix Decompositions | Apr 13, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 |
### [Square Matrix](https://texonom.com/square-matrix-21e528744f2e4d81910de180aa0157ce) A and Q is [Orthogonal Matrix](https://texonom.com/orthogonal-matrix-5e743532b7ac4364b6c1737e2da53798)
R is [Upper Triangular Matrix](https://texonom.com/upper-triangular-matrix-9306a963fe274b20b34438719d709f2f)
$$A = QR$$
1. [Gram-Schmidt Process](https://texonom.com/gram-schmidt-process-60e50ee9d9f042b986cafaf3e0e960b3) โs output is Q
2. $Q^TA = R$
| b2e297ba8eda4bbaa7eb8761a2ab7cff |
|
SVD | null | null | null | null | null | null |
## Singular Value Decomposition (Full SVD)
$$A = U\Sigma V^T$$
> - U is *ortho-normal and left* singular vector (mxm)
> - eigenvectors of $AA^T$
> - $\Sigma$ is diagonal and singular values (mxn)
> - V is *ortho-normal and right *singular vector (nxn)
> - eigenvectors of $A^TA$
>
### SVD Notion
|Title|
|:-:|
|[Singular Vector](https://texonom.com/singular-vector-85b671e8460749eab2e61f550beb61d0)|
|[Singular Value](https://texonom.com/singular-value-651478a593d742fab0c0858d36e8b140)|
|[SVD Reconstruction](https://texonom.com/svd-reconstruction-b0bb92edc992406db2d328795ff26335)|
|[Reduced SVD](https://texonom.com/reduced-svd-85f68348ca3646b9a0a6e0d15b9054db)|
> [SVD์ PCA, ๊ทธ๋ฆฌ๊ณ ์ ์ฌ์๋ฏธ๋ถ์(LSA) ยท ratsgo's blog](https://ratsgo.github.io/from%20frequency%20to%20semantics/2017/04/06/pcasvdlsa/)
> [ํน์ด๊ฐ ๋ถํด(SVD) - ๊ณต๋์ด์ ์ํ์ ๋ฆฌ๋
ธํธ (Angelo's Math Notes)](https://angeloyeo.github.io/2019/08/01/SVD.html)
| c7cdeb1b6b4747c8825b7e23c325d1ab |
Reduced SVD | SVD Notion | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 |
### Reduced SVD Types
|Title|
|:-:|
|[Thin SVD](https://texonom.com/thin-svd-3c4fa0441c4242a3a8b74889e4166f7d)|
|[Compact SVD](https://texonom.com/compact-svd-579e2e7701e64da0be0cbd4263cb4572)|
|[Truncated SVD](https://texonom.com/truncated-svd-297221a6fc7a43fe94c41378450e3854)|
> [[ML] PCA(์ฃผ์ฑ๋ถ๋ถ์), SVD, LDA by Fisher](https://techblog-history-younghunjo1.tistory.com/66)
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Singular Value | SVD Notion | Jun 5, 2023 | Alan Jo | Alan Jo | Jun 5, 2023 | 651478a593d742fab0c0858d36e8b140 |
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Singular Vector | SVD Notion | Aug 31, 2021 | Alan Jo | Alan Jo | Jun 5, 2023 |
A์ row space์ column space ์ฌ์ด์ ์ ํ ๋ณํ์ ๋ํ ์ ๋ณด
left right ์ค ํ๋์ singular vector์ singular value๋ง์ผ๋ก๋ ๋ง์ ์ ๋ณด๋ฅผ ์ป์ ์ ์๋ค
### Left singular vector
### Right singular vector
$V $์๋๋ผ $V^T$์ธ ์ด์ ๋ ๋ฒกํฐ๋ก ๋ถํดํ๋ฉด ๋ฐฉํฅ์ด ๊ฐ๋ก๋ผ์ ๊ณฑํด์ง๋ ๊ฒ
A์ column space๋ฅผ spanํ๋ ๋ฒกํฐ
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SVD Reconstruction | SVD Notion | Jun 5, 2023 | Alan Jo | Alan Jo | Jun 5, 2023 |
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Compact SVD | Reduced SVD Types | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 | 579e2e7701e64da0be0cbd4263cb4572 |
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Thin SVD | Reduced SVD Types | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 | 3c4fa0441c4242a3a8b74889e4166f7d |
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Truncated SVD | Reduced SVD Types | Jun 11, 2023 | Alan Jo | Alan Jo | Jun 11, 2023 | 297221a6fc7a43fe94c41378450e3854 |
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Matrix Multiplication | Matrix Operaters | Apr 17, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 |
### Anti [Commutative Property](https://texonom.com/commutative-property-216366d94b2d43fb87d11e87cc3a233a)
$$AB \neq BA$$
### Sigma representation
$$A(Bu) = (AB)u = w = Cu$$
$$w_i = \sum_k{A_{ik}\sum_r{B_{kr}u_r}}=\sum_r{(\sum_k{A_{ik}B_{kr}})u_r}=\sum_r{(C_{ir})u_r}$$
ibm researcher Shumuel Winograd proved 6 computation or less is impossible for 3x3 multiplication
### Matrix Multiplication Algorithms
|Title|
|:-:|
|[AlphaTensor](https://texonom.com/alphatensor-c88b0c232de14ec787c29196bdffd2e2)|
|[Strassenโs Algorithm](https://texonom.com/strassens-algorithm-a9c7918cfa654b0a9b3d07b23e34da73)|
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Matrix Square | Matrix Operaters | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 |
1. Matrix multiplication is inner product of a row ofย **A **with a column ofย **A**T
2. But the columns ofย **A**Tย are the rows ofย **A**, so the entry corresponds to the inner product of tworows ofย **A** Ifย *pij *ย is the entry of the product
3. So it is symmatric
$$A^2 = A^TA$$
[Symmetric Matrix](https://texonom.com/symmetric-matrix-8ab18b2c502545e08445c0110a090131) also
[Symmetric positive semi-definite](https://texonom.com/symmetric-positive-semi-definite-16ce5c2766e94f3d803ea6ddf2742660)
> [Is a matrix multiplied with its transpose something special?](https://math.stackexchange.com/questions/158219/is-a-matrix-multiplied-with-its-transpose-something-special)
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Matrix Trace | Matrix Operaters | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 |
### ๋๊ฐํฉ
| 0b7fd1b5735549f782a13909912b02a9 |
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AlphaTensor | Matrix Multiplication Algorithms | May 23, 2023 | Alan Jo | Alan Jo | May 23, 2023 | [Tensor](https://texonom.com/tensor-3fb2021969c44483945bacdae2904688) โฃ |
> [Discovering novel algorithms with AlphaTensor](https://www.deepmind.com/blog/discovering-novel-algorithms-with-alphatensor)
> [DeepMind unveils first AI to discover faster matrix multiplication algorithms](https://venturebeat.com/ai/deepmind-unveils-first-ai-to-discover-faster-matrix-multiplication-algorithms/)
> [How a Computer Broke a 50-Year Math Record](https://www.youtube.com/watch?v=fDAPJ7rvcUw)
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Strassenโs Algorithm | Matrix Multiplication Algorithms | May 23, 2023 | Alan Jo | Alan Jo | Sep 12, 2023 | [Divide and Conquer paradigm](https://texonom.com/divide-and-conquer-paradigm-9b10e85e70214b9b8f0bd293f5c297bd) |
only 8 to 7 multiplication for 2x2 multiplication so $\Theta(n^{lg7})$
$$T(n) = 7T(n/2) + \Theta(n^2)$$
| a9c7918cfa654b0a9b3d07b23e34da73 |
Gram-Schmidt Process | Orthogonal Matrix Usages | Apr 6, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 |
๋ฒกํฐ๋ค์ ์งํฉ์ ์๋ก ์ง๊ตํ๋ ๋ฒกํฐ๋ค์ ์งํฉ์ผ๋ก ๋ณํํ๋ ๊ณผ์
๋ชจ๋ ์
๋ ฅ ๋ฒกํฐ๋ค์ด ์ ํ ๋
๋ฆฝ์ด์ด์ผํจ
1. ๋ฒกํฐ ์งํฉ์ผ๋ก ๋ณํ
2. ์ฒซ ๋ฒกํฐ๋ฅผ ๊ทธ๋๋ก ๋๊ณ ๋๋ฒ์งธ ๋ฒกํฐ์์ ์ฒซ๋ฒ์งธ ์ฑ๋ถ ์ ๊ฑฐ (์ง๊ตํ๊ฒ)
3. ๊ณ์ ์์ ๊ฒ๋ค๊ณผ ์ง๊ตํ๊ฒ ์ ๊ฑฐ
4. ๋ชจ๋ ์ป์ ๋ค์ ๋ชจ๋ ์ ๊ทํ
output์ ์ด๋ค์ด ๋ชจ๋ ์ง๊ตํ๋ ํ๋ ฌ์ด๋ฉด์ [Upper Triangular Matrix](https://texonom.com/upper-triangular-matrix-9306a963fe274b20b34438719d709f2f)
์ค์ ๊ณ์ฐ์์๋ ์์น์ ์์ ์ฑ(numerical stability) ๋ฌธ์ ๋ก ์ธํด ์ ํํ ์์ผ๊ฐํ๋ ฌ์ ์ป์ง ๋ชปํ ์
์ด๋ฅผ ํด๊ฒฐํ๊ธฐ ์ํด์ modified Gram-Schmidt
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Determinant | Square Matrix Usages | Oct 4, 2020 | Alan Jo | Alan Jo | Jun 25, 2023 |
## ํ๋ ฌ์
Square Matrix ์ ํ๋์ ์๋ฅผ ๋์
### Determinant Usages
|Title|
|:-:|
|[Geometric mean of Determinant](https://texonom.com/geometric-mean-of-determinant-b907c103917a4e7faddffaa579fe0d08)|
> [[Linear Algebra] Lecture 18 ํ๋ ฌ์์ ํน์ฑ(Properties of Determinants)](https://twlab.tistory.com/40?category=668741)
| 92e4c12610184c8e91a869761d7a5c8c |
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Geometric mean of Determinant | Determinant Usages | Oct 4, 2020 | Alan Jo | Alan Jo | Jun 25, 2023 |
### Determinant and Area
Area of parallelogram by determinant
> [[Linear Algebra] Lecture 20-(2) ํ๋ ฌ์(Determinant)์ ๊ธฐํํ์ ํด์(Geometrical Analysis)](https://twlab.tistory.com/m/44)
| b907c103917a4e7faddffaa579fe0d08 |
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**Circulant matrix** | Square Matrixes | May 13, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 | 0fcb400b1bd84d37876a8e1340a25176 |
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Diagonal Matrix | Square Matrixes | Apr 6, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 |
> [Diagonal matrix](https://en.wikipedia.org/wiki/Diagonal_matrix)
| 622ba1ae252f4e3fbe1328e155430a64 |
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Identity matrix | null | null | null | null | null | null | 30db8dd3fc3b4e7da2d12cbcfc68a25b |
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Symmetric Matrix | Square Matrixes | Apr 17, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 |
### Symmetric Matrixes
|Title|
|:-:|
|[Hermitian matrix](https://texonom.com/hermitian-matrix-15f653dc272b4438bcd9057ea60c1e7f)|
| 8ab18b2c502545e08445c0110a090131 |
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Toeplitz Matrix | Square Matrixes | Apr 19, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 |
๋๊ฐ์ ์์ ์ฑ๋ถ๋ค์ด ๊ฐ์ ์ ์ฌ๊ฐ ํ๋ ฌ
| 7d66a501e56c4cc8b407dc9a586724c4 |
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Hermitian matrix | Symmetric Matrixes | Jun 25, 2023 | Alan Jo | Alan Jo | Jun 25, 2023 | [Hermitian function](https://texonom.com/hermitian-function-af9c5415235f4d3da67d4e0e15389cff) [Hermitian manifold](https://texonom.com/hermitian-manifold-4d106bc7abe4467eb63d09e2189ca08b) |
### self-adjoint matrix
์ค์ ๋์นญ ํ๋ ฌ์ ์ผ๋ฐํ
| 15f653dc272b4438bcd9057ea60c1e7f |
Lower Triangular Matrix | Triangular Matrixes | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 | 68d11496d8614893ab3f3fcedf150a23 |
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Upper Triangular Matrix | Triangular Matrixes | Apr 17, 2023 | Alan Jo | Alan Jo | Apr 17, 2023 | 9306a963fe274b20b34438719d709f2f |
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Correlation Matrix | null | null | null | null | null | null | fefce53c8a2c4d91b111167e7d0e4094 |
|
Covariance Matrix | null | null | null | null | null | null |
$$S = \frac{1}{N}\Sigma_n(x_n - \bar{x})^T(x_n - \bar{x}) = \frac{1}{N}X^TX$$
๊ฐ ํ๊ณผ ์ด์ ๊ทธ ํ๊ณผ ๊ทธ ์ด์ covariance์ด๊ณ ๋๊ฐ์ ์ variance
> [์ฃผ์ฑ๋ถ ๋ถ์(PCA) - ๊ณต๋์ด์ ์ํ์ ๋ฆฌ๋
ธํธ (Angelo's Math Notes)](https://angeloyeo.github.io/2019/07/27/PCA.html)
> [Covariance matrix (๊ณต๋ถ์ฐ ํ๋ ฌ)](https://yonghyuc.wordpress.com/2019/07/13/covariance-matrix-๊ณต๋ถ์ฐ-ํ๋ ฌ/)
> [Covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix)
| 9067b8dc2bbc43ce915c4107eaf84bb5 |
***Direction Cosine Matrix*** | Matrix Usages | Jan 20, 2022 | Alan Jo | Alan Jo | Apr 17, 2023 |
## DCM
๋ฐฉํฅ ์ฝ์ฌ์ธ ํ๋ ฌ
ํ์ ํ๋ ฌ
| 0622b233d72a4ad68a923fe7f8a806be |
Subsets and Splits