Symmetric Matrix

A symmetric matrix is a square matrix when it is equal to its transpose of a matrix. Only a square matrix is symmetric because in linear algebra equal matrices have equal dimensions. Generally, symmetric matrix definition is given by

A = AT

If aij denotes the entries in an i-th row and j-th column, then the symmetric matrix is represented as

aij = aji

Where all the entries of a symmetric matrix are symmetric with respect to the main diagonal. The symmetric matrix examples are given below :

2 x 2 square matrix : \(A = \begin{pmatrix} 4 & -1\\ -1& 9 \end{pmatrix}\)

3 x 3 square matrix : \(B = \begin{pmatrix} 2 & 7 & 3 \\ 7& 9 &4 \\ 3 & 4 &7 \end{pmatrix}\)

What is the Transpose of a Matrix?

A matrix “M” is said to be the transpose of a matrix if the rows and columns of a matrix are interchanged. In this case, the first row becomes the first column, the second row becomes the second column and so on. The transpose of a matrix is given as “MT “.

Consider the above matrix A and B

\(A = \begin{pmatrix} 4 & -1\\ -1& 9 \end{pmatrix}\) ; \(B = \begin{pmatrix} 2 & 7 & 3 \\ 7& 9 &4 \\ 3 & 4 &7 \end{pmatrix}\)

Then, the transpose of a matrix is given by

\(A^{T} = \begin{pmatrix} 4 & -1\\ -1& 9 \end{pmatrix}\) ; \(B^{T} = \begin{pmatrix} 2 & 7 & 3 \\ 7& 9 &4 \\ 3 & 4 &7 \end{pmatrix}\)

When you observe the above matrices, the matrix is equal to its transpose.

Therefore, the symmetric matrix is written as

A = AT and B = BT

Properties of Symmetric Matrix

A symmetric matrix is used in many applications because of its properties. Some of the symmetric matrix properties are given below :

  • A symmetric matrix should be a square matrix.
  • The eigenvalue of the symmetric matrix should be a real number.
  • If the matrix is invertible, then the inverse matrix is a symmetric matrix
  • The matrix inverse is equal to the inverse of a transpose matrix.
  • If A and B be a symmetric matrix which is of equal size, then the summation (A+B) and subtraction(A-B) of the symmetric matrix is also a symmetric matrix.
  • A scalar multiple of a symmetric matrix is also a symmetric matrix.
  • If the symmetric matrix has distinct eigenvalues, then the matrix can be transformed into a diagonal matrix. In other words, it is always diagonalizable.
  • For every distinct eigenvalue, eigenvectors are orthogonal.

Sample Problem

Question :

Show that the product ATA is always a symmetric matrix.

Solution :

Consider a matrix, \(A = \begin{pmatrix} 1 & 2 &3 \\ 4&5 & 6 \end{pmatrix}\)

Now take the transpose of a matrix A,

\(A^{T} =\begin{pmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{pmatrix}\)

Therefore,

ATA = \(\begin{pmatrix} 1 & 2 &3 \\ 4&5 & 6 \end{pmatrix}\)\(\begin{pmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{pmatrix}\)

ATA = \(\begin{pmatrix} 1+4+9 & 4+10+18\\ 4+10+18 & 16+25+36 \end{pmatrix}\)

ATA = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}\)

To prove : The product of ATA is always a symmetric matrix.

So, taking the transpose of ATA ,

(ATA)T = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}^{T}\)

(ATA)T = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}\)

The transpose of ATA is a symmetric matrix.

Hence Proved.

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