Define Symmetric and Skew Symmetric Matrix with Example

Define Symmetric Matrix with Examples

In linear algebra, A square matrix A is called a symmetric matrix if we take a transpose of a matrix and the answer is itself matrix.

such that Symmetric Matrix

Symmetric Matrix Example 3×3

A^t = A

Symmetric Matrix
Symmetric Matrix

A symmetric matrix is always a square matrix because equal matrices have equal dimension. In the symmetric matrix, the number of rows and the number of columns are the same as it is a square matrix. Every real symmetric matrix can be diagonalizable.

General form of Symmetric matrix

A symmetric matrix is important and useful in many applications because of its application. Some examples of well-known symmetric matrices are the correlation matrix covariance matrix and distance matrix.

Before we move further first we discuss that what is a square matrix is and what is the transpose of a matrix

A matrix is said to be a square matrix if and only if the number of columns of rows is equal to the number of columns. A number of rows shown by m and number of columns are shown by n. in square matrix m = n.

Transpose of a symmetric matrix

If we want to take the transpose of any matrix then we interchanging the rows and columns of the original matrix. If a matrix A has m × n order and we take transpose then the order is changing and the order is n × m.

Example of symmetric matrix

A=\;\begin{bmatrix}1&2\\2&1\end{bmatrix}

order of this matrix A is 2 by 2

Firstly we change the first row to the first column

Then we change the second row to the second column

Example of a symmetric matrices

A=\;\begin{bmatrix}3&-2&4\\-2&6&2\\4&2&3\end{bmatrix}

Here the order of the matrix A is 3 -by 3.

Properties of symmetric matrices

  • The product of two symmetric matrices may not be symmetric. That is if

A^t = A , B^t = B then (AB)^t = B^t A^t = BA not equal to AB is general.it is only possible when A and B are commute like AB = BA

  • The sum of two symmetric matrices is again a symmetric matrices.
  • The difference of two symmetric matrices is also again a symmetric matrice.
  • The most important property of symmetric matrices is that their eigenvalues behave very nicely.
  • Hermitian matrice is also symmetric matrix if the entries are complex number.
  • If all eigenvalues of symmetric matrix A are different  then the  matrix A can transformed into its diagonal matrice.
  • There is no complex number in Eigen value of a symmetric matrix that is means there are all real numbers.
  • Symmetric matrices has linearly independent eigen vector.

Note symmetric matrix

The zero matrix is also a symmetric matrix.

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