A matrix is called a symmetric matrix if its transpose is equal to the matrix itself. Only a square matrix is symmetric because in linear algebra equal matrices have equal dimensions.

How do you know if a matrix is symmetric? Generally, the symmetric matrix is defined as

A = A^T

Where A is any matrix, and A^T  is its transpose.

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

a_ij = a_ji

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, and the second row becomes the second column and so on. The transpose of a matrix is given as “M^{T “}.

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 = A^T and B = B^T

Symmetric Matrix Inverse

Since the symmetric matrix is taken as A, the inverse symmetric matrix is written as A^-1, such that it becomes

A × A^-1 = I

Where “I” is the identity matrix.

If a matrix contains the inverse, then it is known as invertible matrix, and if the inverse of a matrix does not exist, then it is called a non-invertible matrix.

The symmetric matrix inverse can be found using two methods. They are

• Adjoint Method
• Gauss-Jordan Elimination method.

It is noted that inverse of the given symmetric matrix is also a symmetric matrix.

Symmetric Matrix Determinant

Finding the determinant of a symmetric matrix is similar to find the determinant of the square matrix. A determinant is a real number or a scalar value associated with every square matrix. Let A be the symmetric matrix, and the determinant is denoted as “det A” or |A|. Here, it refers to the determinant of the matrix A. After some linear transformations specified by the matrix, the determinant of the symmetric matrix is determined.

Read More on Determinant Of A Matrix

Properties of Symmetric Matrix

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

• The 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.

Symmetric and Skew Symmetric Matrix

A matrix is Symmetric Matrix if transpose of a matrix is matrix itself.

Consider a matrix A, then

Transpose of A = A

A matrix is Skew Symmetric Matrix if transpose of a matrix is negative of itself.

Consider a matrix A, then

Transpose of A = – A

Read More on Symmetric Matrix And Skew Symmetric Matrix

Sample Problem

Question :

Show that the product A^TA 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,

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

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

A^TA = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}\)

To prove : The product of A^TA is always a symmetric matrix.

So, taking the transpose of A^TA ,

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

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

The transpose of A^TA is a symmetric matrix.

Hence Proved.

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