Symmetric Matrix and Skew Symmetric Matrix

Symmetric Matrix

As we know that similar matrices have similar dimensions, therefore only the square matrices can either be symmetric or skew-symmetric. In other words, we can say that both a symmetric matrix and a skew-symmetric matrix are square matrices. The difference between both symmetric matrix and a skew-symmetric matrix is that symmetric matrix is always equivalent to its transpose whereas skew-symmetric matrix is a matrix whose transpose is always equivalent to its negative. For example, If M is a symmetric matrix then M = MT and if M is a skew-symmetric matrix then M = - MT   

The sum of symmetric matrix and skew-symmetric matrix is always a square matrix. Below mentioned formula will be used to find the sum of the symmetric matrix and skew-symmetric matrix. For example,

Let M be the square matrix then,

  • M = (½) × ( M + M’) + (½) ×( M - M’)

  • M’ is the transpose of a matrix.

  • 1/2( M + M’) is a symmetric matrix

  • 1/2( M - M’) is a skew-symmetric matrix

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Symmetric Matrix Meaning

A matrix is onlys stated as a symmetric matrix if its transpose is equivalent to the matrix itself. Only a square matrix is a symmetric matrix because in linear algebra similar matrices have similar dimensions. How will you find whether the matrix given is a symmetric matrix or not ?

Generally, symmetric matrix is expressed as

M= MT            

Where M is any matrix and MT is the transpose of matrix.

If aij represents any elements in an ith column and jth rows, then symmetric matrix is expressed as 

aij = aji

Where each element of a symmetric matrix is symmetric in terms to the main diagonal. Let us understand the concept of symmetric matrix through the symmetric matrix example given below.


Symmetric Matrix Example

The below symmetric example helps you to clearly understand the concept of skew matrix

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In the above symmetric matrix example given below, we can see aij =aji  for all the values of i and j. Here a12 =  a21 = 3, = -a13 =  a31 = 8 a23 =  a32  = -4.In other words, it is stated that the transpose of matrix M is equivalent to the matrix itself (M=MT )which implies that matrix M is symmetric.


What Is a Skew-Symmetric Matrix With an Example?

A square Matrix A is defined as  skew-symmetric if aij = aji for all the values of i and j. In other words, we can say  that matrix P is said to be skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A i.e (AT = −A). Also, it is important to note that all the elements present in the main diagonal of the skew-symmetric matrix are always zero. Let us understand this through a skew-symmetric matrix example.


Skew-Symmetric Matrix Example

The below skew- symmetric example helps you to clearly understand the concept of skew matrix.

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In the above skew matrix symmetric example, we can see all the elements present in the main diagonal of matrices A are zero and also a12 = -2 and  a21 = -2 which implies that a12 = a21 .This conditions is valid for each values of i and j.


Properties of Skew-Symmetric Matrix

Some of the properties of skew-symmetric matrix examples are given below:

  • When two skew-matrices are added, then the resultant matrix will always be a skew-matrix.

  • The result of the scalar product of skew-symmetric matrices is always a skew-symmetric matrix.

  • All the elements included in the main diagonal of the skew matrix are always equal to zero. Hence, the total of all the elements of the skew matrix in the main diagonal is zero.

  • When both identity matrix and skew-symmetric matrix are added, the matrix obtained is invertible.

  • The determinants of skew-symmetric matrices are always non-negative.


Solved Example

  1. For the Given Below Matrix M, Verify That (M + M') Is a Symmetric Matrix.

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Solution: 

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As, (M + M') = M + M'

Hence, (M + M') is a symmetric matrix.


  1. Show That Matrix M Given Below is a Skew- Symmetric Matrix.

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Solution:

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∴, M = M’

Hence, M is a skew-symmetric matrix.


Quiz Time

  1. If Matrix M Is Both a Symmetric Matrix and Skew-Symmetric Matrix Then Matrix M is

    1. A scalar matrix

    2. A diagonal matrix

    3. A zero matrix of order m * m

    4. A rectangular matrix


2. The Diagonal Entities of a Skew-Symmetric Matrix Are

    1. All zeroes

    2. Can be any number

    3. Are all equals to similar scalar k (0

    4. None of these

FAQ (Frequently Asked Questions)

1. What Are the Properties of a Symmetric Matrix?

There are multiple applications of symmetric matrices due to its properties. Some of the symmetric matrix properties are mentioned below:

  • It is necessary for a symmetric matrix to be a square matrix.

  • The eigenvalue of the symmetric matrix should always be given in a real number.

  • If the matrix given is invertible, then the inverse matrix will be considered a symmetric matrix

  • The inverse matrix will always be equivalent to the inverse of a transpose matrix.

  • If P and Q are symmetric matrices of equal size, then the total of (P + Q) and subtraction of (P- Q) of the symmetric matrix will also be the symmetric matrix.

  • A scalar multiple of a symmetric matrix will also be considered as a symmetric matrix.

  • If the symmetric matrix has different eigenvalues, then the matrix can be changed into a diagonal matrix. In other words, a symmetric matrix is always diagonalizable.

  • Eigenvectors are orthogonal for every different eigenvalue,

2.  What Are the Determinants of a Skew-Symmetric Matrix?

If M is a skew-symmetric matrix, which is also considered as a square matrix,  then the determinant of M should satisfy the below-given situation:

Det (MT) = det (-M) = (-1)n det(M)

The inverse of skew-symmetric matrix is not possible as the determinant of it having odd order is zero and therefore it is singular.

The determinants of a skew-symmetric matrix is also one of the  properties of  skew-symmetric matrices. If we have any  skew-symmetric matrix with odd order then we can straightly write its determinants equals to zero. The property of the determinants of a skew-symmetric matrix can be verified using an example of 3 by 3 matrix. The determinants of a skew matrix can be found out using co-factors and can state that its determinant is equivalent to zero.