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 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 : \(B = \begin{pmatrix} 2 & 7 & 3 \\ 7& 9 &4 \\ 3 & 4 &7 \end{pmatrix}\)
Symmetric Matrix Inverse
Since the symmetric matrix is taken as A, the symmetric matrx is written as A-1, such that it becomes
A × A-1 = 1
Where I is known as 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, 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.
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 “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}
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 A^{T}A 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^{T}A = \(\begin{pmatrix} 1 & 2 &3 \\ 4&5 & 6 \end{pmatrix}\)\(\begin{pmatrix} 1 & 4\\ 2 & 5\\ 3 & 6 \end{pmatrix}\)
A^{T}A = \(\begin{pmatrix} 1+4+9 & 4+10+18\\ 4+10+18 & 16+25+36 \end{pmatrix}\)
A^{T}A = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}\)
To prove : The product of A^{T}A is always a symmetric matrix.
So, taking the transpose of A^{T}A ,
(A^{T}A)^{T} = \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}^{T}\)
(A^{T}A)^{T }= \(\begin{pmatrix} 14 & 32\\ 32 & 77 \end{pmatrix}\)
The transpose of A^{T}A is a symmetric matrix.
Hence Proved.
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Symmetric Matrix & Skew Symmetric Matrix | Types Of Matrices |
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