List of Linear Matrix Algebra Formulas Latex Code

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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. Matrix Algebra

• 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 j-th row i-th column element of matrix A_{ij} is the i-th row j-th 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 ij-th 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 i-th row and j-th 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| = |A||B|...|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