Symmetric Matrix

By Kardi Teknomo, PhD .
LinearAlgebra

< Next | Previous | Index >

Symmetric Matrix

A square matrix Symmetric Matrix is symmetric if its transpose is equal to itself, that is Symmetric Matrix

Symmetric matrix is important in many applications because of its properties. Examples of well known symmetric matrices are correlation matrix, covariance matrix and distance matrix.

You can easily create symmetric matrix either by

  1. Multiplying a matrix by its transpose: if Symmetric Matrix is a rectangular matrix, then Symmetric Matrix and Symmetric Matrix are symmetric matrices.
  2. Adding a matrix by its transpose: if Symmetric Matrix is a square matrix, then Symmetric Matrix is a symmetric matrix.

The interactive program below is designed to answers the question whether the given input matrix is a symmetric matrix. When you click Random Example button, it will create random input matrix to provide you with many examples of symmetric and non-symmetric matrices. You may also want to try to type your own input matrix to test whether it is a symmetric matrix.

Properties

Some important properties of symmetric matrix are

  • Symmetric matrix is always a square matrix
  • If Symmetric Matrix is a symmetric matrix order Symmetric Matrix with real entries then
    o The transpose matrix Symmetric Matrix is also a symmetric matrix
    o Scalar multiple of the matrix Symmetric Matrix is also a symmetric matrix
    o The inverse matrix Symmetric Matrix is also a symmetric matrix, if it is invertible.
    o The inverse of the transpose matrix is equal to the matrix inverse Symmetric Matrix
    o A half of the summation to its transpose produces the matrix itself, Symmetric Matrix
    o Subtraction to its transpose produces null matrix, Symmetric Matrix
    o Symmetric matrix Symmetric Matrix has linearly independent Eigen vectors.
    o The Eigen values of symmetric matrix Symmetric Matrix are all real numbers (no complex numbers).
    o If all eigenvalues of symmetric matrix Symmetric Matrix are distinct (no multiple Eigen values), then matrix A can be transformed into diagonal matrix
    o Eigenvectors of distinct eigenvalues are orthogonal.
    o The number of non-zero eigenvalues is equal to its rank Symmetric Matrix .
    o There is an orthogonal matrix Symmetric Matrix that diagonalizes symmetric matrix Symmetric Matrix by Symmetric Matrix (spectral decomposition).
  • If Symmetric Matrix and Symmetric Matrix are symmetric matrices of the same size, then
    o The summation of the matrix Symmetric Matrix is also a symmetric matrix
    o The subtraction of the matrix Symmetric Matrix is also a symmetric matrix, if Symmetric Matrix
  • Let Symmetric Matrix be any rectangular matrix size Symmetric Matrix by Symmetric Matrix such that Symmetric Matrix , then we can form symmetric matrix Symmetric Matrix and Symmetric Matrix . The non-zero eigenvalues of Symmetric Matrix and Symmetric Matrix are equal. The rank of both matrices are equal, Symmetric Matrix .
  • Hermitian matrix Symmetric Matrix is a symmetric matrix with entries of complex number (across the diagonal, the entries are complex conjugate).
  • Both Hermitian and Unitary matrix (including symmetric and orthogonal matrix) are called normal matrix because the Eigen vectors form orthonormal set.

See also : Singular Value Decomposition , orthogonal vector , orthogonal matrix , matrix rank , Spectral Decomposition , Symmetric matrix using MS Excel

< Next | Previous | Index >

Rate this tutorial or give your comments about this tutorial

This tutorial is copyrighted .