Definition of a Symmetric Matrix

A square matrix A is symmetric if and only if A = A^T where A^T is the transpose of matrix A .
A symmetric matrix may be recognized visually: The entries that are symmetrically positioned with respect to the main diagonal are equal as shown in the example below of a symmetric matrix.

These are examples of symmetric matrices.
a)

By simple inspection, the entries that are symmetrically positioned with respect to the main diagonal are both equal to -2 and therefore matrix A is symmetric.
Also, the transpose of matrix A is obtained by interchanging the row of the matrix into a column. The transpose of matrix A is
Note that A^T = A and therefore matrix A is symmetric.

b)

The transpose of matrix B is obtained by interchanging the column of the matrix into a row. The transpose of matrix B is
Note that B^T = B and therefore matrix B is symmetric.

The Product of Any Matrix and it Transpose is Symmetric

For any matrix \( A \) of size \( m \times n\) , the matrices \( A A^T \) and \( A^T A \) are symmetric and have sizes \( m \times m \) and \( n \times n \) respectively.
Let \( A = \begin{bmatrix} -3 & -3 & 9\\ 2 & -7 & 1 \end{bmatrix} \) is a \( 2 \times 3 \) matrix

\( A^T = \begin{bmatrix} -3 & 2\\ -3 & -7 \\ 9 & 1 \end{bmatrix} \)

\( A A^T = \begin{bmatrix} -3 & -3 & 9\\ 2 & -7 & 1 \end{bmatrix} \begin{bmatrix} -3 & 2\\ -3 & -7 \\ 9 & 1 \end{bmatrix} \)
\( \quad \quad = \begin{bmatrix} 99 & 24\\ 24 & 54 \end{bmatrix} \)

Hence \( A A^T \) is a \( 2 \times 2 \) symmetric matrix.

\( A^T A = \begin{bmatrix} -3 & 2\\ -3 & -7 \\ 9 & 1 \end{bmatrix} \begin{bmatrix} -3 & -3 & 9\\ 2 & -7 & 1 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 13 & -5 & -25\\ -5 & 58 & -34\\ -25 & -34 & 82 \end{bmatrix} \) is a \( 3 \times 3 \)

Hence \( A^T A \) is a \( 3 \times 3 \) symmetric matrix.

Properties of the Symmetric Matrices

Some of the most important properties of the symmetric matrices are given below.

1.    If \( A \) is a symmetric matrix, then \( A^T = A \), where\( A^T \) is the transpose of matrix \( A \).
2.    If \( A \) and \( B \) are symmetric matrices, then \( A \pm B \) are also symmetric.
3.    If \( A \) and \( B \) are symmetric matrices, then \( AB + BA \) is also symmetric. (\( A \) and \( B \) must be of the same size)
4.    If \( A \) is a symmetric matrix, then its inverse matrix \( A^{-1} \) ,if it exists ,is also symmetric.
5.    If \( A \) is a symmetric matrix, then \( k A \) is also symmetric; where \( k \) is any real number.
6.    If \( A \) is a symmetric matrix, then \( A^n \) is also symmetric; \( n \) is any positive integer.
7.    For any matrix \( B \), the matrices \( B B^T \) and \( B^T B \) are symmetric.

Examples with Solutions

Example 1
Which if the following matrices are symmetric?
a) \( A = \begin{bmatrix} 0 & -1 & 1 \\ -1 & 0 & 0 \end{bmatrix} \)      b) \( B = \begin{bmatrix} -2 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{bmatrix} \)      c) \( C = \begin{bmatrix} 5 & -7 & 0 & 0\\ 5 & -7 & 0 & 0\\ 5 & -7 & 0 & 0\\ 5 & -7 & 0 & 0 \end{bmatrix} \)      d) \( D = \begin{bmatrix} 1 & 4 & 0 & 9\\ 4 & 2 & -4 & 0\\ 0 & -4 & 4 & 0\\ 9 & 0 & 0 & -5 \end{bmatrix} \)


Solution
Matrices \( B \) and \( D \) are symmetric.

Example 2
Symmetric matrices \( A \) and \( B \) are given by \( A = \begin{bmatrix} -1 & 1 \\ 1 & 2 \end{bmatrix} \) , \( B = \begin{bmatrix} 4 & 2 \\ 2 & -3 \end{bmatrix} \).
Show that \( A+B \) and \( A-B \) are symmetric. (Verify property 2)

Solution
Calculate \( A + B \)
\( A + B = \begin{bmatrix} -1 & 1 \\ 1 & 2 \end{bmatrix} + \begin{bmatrix} 4 & 2 \\ 2 & -3 \end{bmatrix} \)
\( \quad = \begin{bmatrix} 3 & 3 \\ 3 & -1 \end{bmatrix} \)

Calculate \( A - B \)
\( A - B = \begin{bmatrix} -1 & 1 \\ 1 & 2 \end{bmatrix} - \begin{bmatrix} 4 & 2 \\ 2 & -3 \end{bmatrix} \)
\( \quad = \begin{bmatrix} -5 & -1 \\ -1 & 5 \end{bmatrix} \)

Hence \( A + B \) and \( A - B \) are also symmetric matrices.

Example 3
Find the inverse of the symmetric matrix \( A = \begin{bmatrix} -1 & -2\\ -2 & 0 \end{bmatrix} \) and show that this inverse matrix is symmetric.(Verify properties 4)

Solution
Use the formula of the inverse of a \( 2 \times 2 \) matrix \( \begin{bmatrix} x & y \\ z & w \\ \end{bmatrix}^{-1} = \dfrac{1}{xw - yz} \begin{bmatrix} w & - y \\ - z & x \\ \end{bmatrix} \) to find \( A^{-1} \)
\( A^{-1} = \dfrac{1}{-4} \begin{bmatrix} 0 & 2\\ 2 & -1 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 0 & -\dfrac{1}{2}\\ -\dfrac{1}{2} & \dfrac{1}{4} \end{bmatrix} \)
Hence \( A^{-1} \) is also symmetric.

Example 4
Let matrix \( A = \begin{bmatrix} -3 & 1\\ 1 & 7 \end{bmatrix} \)
Calculate \( A^3 \) and verify that it is symmetric. (Verify properties 6)

Solution
\( A^3 = \begin{bmatrix} -3 & 1\\ 1 & 7 \end{bmatrix} \begin{bmatrix} -3 & 1\\ 1 & 7 \end{bmatrix} \begin{bmatrix} -3 & 1\\ 1 & 7 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 10 & 4\\ 4 & 50 \end{bmatrix} \begin{bmatrix} -3 & 1\\ 1 & 7 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} -26 & 38\\ 38 & 354 \end{bmatrix} \)
Hence \( A^3 \) is a symmetric matrix.

Example 5
Find the real numbers \(a \), \( b \) and \( c \) so that the matrix \( A = \begin{bmatrix} 0 & a+b & c+2 \\ a & 2 & c \\ 4 & a+b & 4 \end{bmatrix} \) is symmetric.

Solution
For the given matrix to be symmetric, we have to have the following equations satisfied simultaneously:
\( a = a + b \) , \( c + 2 = 4 \) , \( c = a + b \)
Solve the system of equations in \( a,\; b,\; c \) above to obtain
\( b = 0 , \; c = 2 , \; a = 2 \) which are the values so that the given matrix is symmetric.

Example 6
Given the symmetric matrices \( A = \begin{bmatrix} -1 & 3 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 8 \\ 8 & 9 \end{bmatrix} \), verify that \( AB + BA \) is also symmetric (Property 3 above)

Solution
Calculate \( AB \)
\( AB = \begin{bmatrix} -1 & 3 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 4 & 8 \\ 8 & 9 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 20 & 19 \\ 44 & 60 \end{bmatrix} \)

Calculate \( BA \)
\( BA = \begin{bmatrix} 4 & 8 \\ 8 & 9 \end{bmatrix} \begin{bmatrix} -1 & 3 \\ 3 & 4 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 20 & 44\\ 19 & 60 \end{bmatrix} \)

Calculate \( AB + BA \)
\( AB + BA = \begin{bmatrix} 20 & 19 \\ 44 & 60 \end{bmatrix} + \begin{bmatrix} 20 & 44\\ 19 & 60 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 40 & 63\\ 63 & 120 \end{bmatrix} \)
Hence \( AB + BA \) is a symmetric matrix.

Questions (with solutions given below)

• Part 1
  Matrices \( A \) and \( B \) are such that \( A B = B A = I \) where \( I \) is the identity matrix . Use any of the properties above to explain that if one of the matrices is symmetric, then the other one is also symmetric.
  
• Part 2
  Given the matrices \( A = \begin{bmatrix} -5 & -2 \\ - 1 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 1 \\ 0 & 4 \end{bmatrix} \).
  Calculate \( A + B \) and explain why \( (A + B)^T = A + B \) without any further computations.
  
• Part 3
  Let matrix \( A = \begin{bmatrix} - 1 & - b\\ c & c^2 \end{bmatrix} \). Find the real numbers \( b \) and \( c \) so that matrix \( A \) is symmetric and \( Det (A) = - 1 \).
  
• Part 4
  Show that any symmetric matrix of the form \( A = \begin{bmatrix} a & \sqrt{1-a^2} \\ \\ \sqrt{1-a^2} & - a \\ \end{bmatrix} \) , with \( |a| \lt 1 \), is such that \( A = A^T = A^{-1} \) and calculate \( A^n \) where \( n \) is a positive integer.
  

Solutions to the Above Questions

• Matrices with Examples and Questions with Solutions.
  
• Transpose of a Matrix.
  
• Determinant of a Square Matrix.
  
• Inverse Matrix Questions with Solutions.
  
• Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres
• Introduction to Linear Algebra - Fifth Edition (2016) - Gilbert Strang
• Linear Algebra Done Right - third edition, 2015 - Sheldon Axler
• Linear Algebra with Applications - 2012 - Gareth Williams