This is the review and cheatsheet of some basic linear algebra knowledge during my undergraduate study.

## 1. Basic

**Determinant**

A scalar value that can be computed from elements of a square matrix.

where $a_{ij}$ is the element of A and $M_{ij}$ is minor.

**Matrix Inverse**

A square matrix M has an inverse that M is invertible if the determinant $|M| \neq 0$.

The inverse properties are

1.

2.

3.

**Orthogonal Matrices**

A square matrix $A \in R^{nn}$ is orthogonal:

**Matrix multiplication**

\(A \in R^{mn}\) and \(B \in R^{np}\)

**Matrix transpose**

**Trace**

The trace of a square matrix $A \in R^{nn}$ is denoted as tr(A), which is the sum of diagonal elements in the matrix:

$tr(A)=\sum_{i=1}^{n}A_{ii}$.

Properties are below:

1.

2.

3.

4.

5.

**Rank**

The column rank of a matrix $A \in R^{mn}$ is the size of the largest subset of columns of A that constitute a linearly independent set. The same definition as for row rank. For any matrix, the column rank is equal to the row rank. Both are denoted as rank(A). Properties are below:

1.

2.

3.

4.

**Eigenvalue and Eigenvectors**

Given a square matrix A, \(\lambda\) is an eigenvalue and x is the corresponding eigenvector if

which equals to

The properties are

1.

2.

3.The rank of A is equal to the number of non-zero eigenvalues of A

4.If A is non-singular then $1/\lambda_{i}$ is an eigenvalue of $A^{-1}$ with associated eigenvector.

5.The eigenvalues of a diagonal matrix are just the diagonal entries.

**The gradient**

$f: R^{mn} \to R$ is a function that input a matrix A and returns a value. The gradient of f is the matrix of partial derivatives.

## 2.Other definitions and calculations

**Laplace Matrix (simple graph)**

Given a simple graph G with n vertices, its Laplace Matrix \(L_{nn}\) is defined as: L=D-A, where D is the degree matrix and A is the adjacency matrix. D is a diagonal matrix which includes the information about the degree of each vertex. A is the adjacency matrix which only includes 1 and 0 since G is a simple graph and the diagonal are all 0.**symmetrix normalized laplacian**

**Singular value decomposition**

Assume $A \in R^{mn}$ and all elements in M belongs to real or plural values. There exists a decomposition that

where U and V are orthogonal that $U^{T}U=I_{mm}$ and $V^{T}V=I_{nn}$.

**Eigen decomposition**

$A \in R^{nn}$ is a square matrix with n linear independent eigenvectors $q_{i}$(i=1…n). A can be factorized as

where Q is the square n by n matrix with ith column is the eigenvector of A, and $\Lambda$ is the diagonal matrix with diagonal elements are the corresponding eigenvalues.