Symmetric and skew-symmetric matrix: examples and properties

Symmetric matrix

Definition of a symmetric matrix: A square matrix A = [a_ij] is called a symmetric matrix if a_ij = a_ij, for all i,j values

Example of the symmetric matrix:

is an example of a symmetric matrix

Note: Matrix A is symmetric if A’ = A (where A’ is the transpose of the matrix)

Transpose of matrices

To understand if a matrix is a symmetric matrix, it is very important to know about the transpose of a matrix and how to find the transpose of a matrix.
If the rows and columns of an m×n matrix are interchanged to get an n × m matrix, the new matrix obtained is called the transpose of the given matrix.

Maths Topics Math Articles Matrices Algebra Types of Matrices

Skew-Symmetric Matrix

Definition of Skew-symmetric matrix: A square matrix A = [aij] is a skew-symmetric matrix if aij = -aji, for all values of i,j.

If we put i=j, then,
a_ii = -a_ii
⇒ 2a_ii = 0
⇒ a_ii = 0

Thus, in a skew-symmetric matrix, all diagonal elements are zero.

Example of a Skew-symmetric matrix:

is an example of skew-symmetric matrices.

Note: A square matrix A is skew-symmetric if A’ = -A.

Properties of Symmetric and skew-symmetric matrices

1. Every square Matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
2. If A is a symmetric Matrix, then Aⁿ, n belonging to the Natural number, will be symmetric.
3. A is skew-symmetric, then Aⁿ, n N will be symmetric if n is even, and Aⁿ will be skew-symmetric if n is odd.
4. Null Matrix is always symmetric and skew-symmetric.
5. If A and B are both symmetric, then AB + BA will be symmetric, and AB – BA will be skew-symmetric.

What is the difference between a symmetric matrix and a skew-symmetric matrix?

A symmetric matrix and a skew-symmetric matrix are the square matrices. But the major difference between them is:

1. The symmetric matrix equals its transpose.
   ⟹If A is a symmetric matrix, then A = A^T
2. Whereas a skew-symmetric matrix is a matrix whose transpose equals its negative.
   ⟹If A is a skew-symmetric matrix, then A^T = – A.

For better understanding of matrices, also read:

FAQs on Symmetric and Skew-symmetric Matrix

Can a matrix be both symmetric and skew-symmetric?

No, a matrix cannot be symmetric and skew-symmetric unless it is the null matrix (a matrix with zero elements). In a non-null matrix, the presence of non-zero diagonal elements in a symmetric matrix contradicts the property of having zero diagonal elements in a skew-symmetric matrix.

Are symmetric and skew-symmetric matrices always square matrices?

Both symmetric and skew-symmetric matrices are defined for square matrices only. In other words, their number of rows equals the number of columns.

Are the eigenvalues of a symmetric matrix real?

Yes, all eigenvalues of a symmetric matrix are real. This property is known as the spectral theorem for symmetric matrices.

What is the relationship between the eigenvectors of a symmetric matrix?

The eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal (perpendicular) to each other.

Can a matrix be both symmetric and diagonal?

A diagonal matrix where all non-diagonal elements are zero is symmetric and diagonal.

Can a matrix be both skew-symmetric and diagonal?

No, a matrix cannot be skew-symmetric and diagonal unless it is the null matrix. In a non-null matrix, the presence of non-zero diagonal elements contradicts the property of having zero diagonal elements in a skew-symmetric matrix.

How can we determine if a matrix is symmetric or skew-symmetric?

To determine if a matrix is symmetric, we compare it to its transpose. If the matrix is equal to its transpose (A = A'), it is symmetric. To determine if a matrix is skew-symmetric, we compare it to the negation of its transpose. If the matrix equals the negation of its transpose (A = -A'), it is skew-symmetric.