List of Linear Matrix Algebra Formulas Latex Code
math_beginner 20230116 #math #matrix #algebra #eigenvalues #determinantsList of Linear Matrix Algebra Formulas Latex Code
Navigation
In this blog, we will summarize the latex code for linear matrix algebra formulas, including matrix multiplication, transpose, inverse matrix, determinants, hermitian matrices, determinants, eigenvalues and eigenvectors, orthogonal matrices, etc.
 1. Matrix Algebra
 1.1 Matrix Products/Multiplication
 1.2 Matrix Transpose
 1.3 Inverse Matrix
 1.4 Determinants
 1.5 Matrix Product Rules
 1.6 Orthogonal Matrices
 1.7 Sets of Linear Simultaneous Equations
 1.8 Hermitian Matrices
 1.9 Eigenvalues and Eigenvectors

1.1 Matrix Products/Multiplication
Equation
Latex Code
(AB)_{ij}=\sum^{l}_{k=1} A_{ik}B_{kj}
Explanation
Assume A is a matrix with shape [n, l] and B is a matrix with shape [l, m]. Then the matrix products or multiplication is defined by the above equations. The latex code is also attached.

1.2 Matrix Transpose
Equation
Latex Code
(A^{T})_{ij} = A_{ji}
Explanation
If A is a matrix with shape [n,l], then transpose matrix A^{T} has shape [l,n]. And the jth row ith column element of matrix A_{ij} is the ith row jth column value of transpose matrix A^{T}

1.3 Inverse Matrix
Equation
Latex Code
A^{1}A=AA^{1}=I \\ A^{1}=\frac{A^{*}}{A} \\ A^{*} = \begin{bmatrix} A_{11} & A_{11} & ... & A_{1n} \\ A_{21} & A_{22} & ... & A_{2n} \\ ... & ... & ... & ... \\ A_{n1} & A_{n2} & ... & A_{nn} \end{bmatrix}^{T}
Explanation
denotes the transpose matrix of cofactor of original element in the ijth position of a_{ij} of original matrix A. The cofactor matrix A_{ij} is defined as (â??1)^{i+j} times the determinant of the matrix A with the ith row and jth column deleted. The matrix product of inverse matrix A^{1} and original matrix A is the identity matrix I. A^{1}A=AA^{1}=I.

1.4 Determinants
Equation
Latex Code
A=\sum_{i,j,k,...}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}...
Explanation
If A is a square matrix then the determinant of A, A or det(A) is defined as above. And the number of the suffixes is equal to the order of the matrix.

1.5 Matrix Product Rules
Equation
Latex Code
(AB...N)^{T} = N^{T}...B^{T}A^{T} \\ (AB...N)^{1} = N^{1}...B^{1}A^{1} \\ AB...N = AB...N
Explanation

1.6 Orthogonal Matrices
Equation
Latex Code
Q^{1} = Q^{T} \\ Q=\pm 1
Explanation
An orthogonal matrix Q is a square matrix whose columns q_{i} form a set of orthonormal vectors. For any orthogonal matrix Q, Q^{T} is also orthogonal. The inverse matrix of Q equals to the transpose matrix Q^{T}.

1.7 Sets of linear simultaneous equations
Equation
Latex Code
Ax=b \\ x=A^{1}b
Explanation
If A is square matrix then Ax = b has a unique solution x = A^{â??1}b if A^{â??1} exists, i.e., if A \ne 0.

1.8 Hermitian Matrices
Equation
Latex Code
A^{+}={(A^{*})}^{T}
Explanation
The Hermitian conjugate of A is A^{+}, which is defined as {(A^{*})}^{T}. A^{*} is a matrix each of whose components is the complex conjugate of the corresponding components of A. If A = A^{+} then A is called a Hermitian matrix.

1.9 Eigenvalues and Eigenvectors
Equation
Latex Code
A\mathbf{u}=\lambda\mathbf{u} \\ P_{n}(\lambda)=A\lambda I
Explanation
The n eigenvalues and eigenvectors of an n by n matrix A are the solutions of the equation . The eigenvalues are the zeros of the polynomial of degree n . Additionally, is called the characteristic equation of the matrix. Trace of matrix A is defined as . Determinants of a matrix A equals to the product of eigenvalues