List of Linear Matrix Algebra Formulas Latex Code

math_beginner 2023-01-16 #math #matrix #algebra #eigenvalues #determinants
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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

    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




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