Symmetric and Skew Symmetric Matrix
The symmetric and skew symmetric matrix has one important characteristic in common; they both are square matrices. Among all the different kinds of matrices, symmetric and skew matrices are some of the most important ones that are used widely in machine learning.
In this article, let's learn about symmetric and skew symmetric matrices, their definitions, properties, and their differences with solved examples.
What Are Symmetric and Skew Symmetric Matrix?
The symmetric and skew symmetric matrix has a close relationship with reach other. This can be understood on learning the definition of each of them. A matrix whose transpose is equal to the matrix itself, is called a symmetric matrix. And a matrix whose transpose is equal to the negative of the given matrix is called a skew symmetric matrix. Here we have considered a matrix A and the basic formula representing a Symmetric and Skew Symmetric Matrix is as follows.
Let us get to know more about the symmetric and skew symmetric matrix.
Symmetric Matrix Definition
A square matrix B which of size n x n is considered to be symmetric if and only if B^{T} = B. Consider the given matrix B, that is, a square matrix that is equal to the transposed form of that matrix, called a symmetric matrix.
This can be represented as: B = \(\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}} \) is the symmetric matrix, then \(b_{ij}\) = \(b_{ji}\) for all i and j or 1 ≤ i ≤ n, and 1 ≤ j ≤ n. Here, n is all the natural numbers. \(b_{ij}\) is an element at position (i, j) which is i^{th} row and j^{th }column in matrix B and \(b_{ji}\) is an element at position (j, i) which is j^{th} row and i^{th} column in matrix B.
Example:
Let’s take an example of a matrix B,
Here, we can see that, B^{T} = B, \(b_{12}\) = \(b_{21}\) =3, and \(b_{13}\) = \(b_{31}\) = 6. Thus, B is a symmetric matrix.Skew Symmetric Matrix Definition
A square matrix B which of size n x n is considered to be skew symmetric if and only if B^{T} = B. That is, a transposed form of a matrix that is equal to the negative of that matrix is called a skew symmetric or antisymmetric matrix. This can be represented as:
If B = \(\left[\mathrm{b}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}} \) is the skew symmetric matrix, then \(b_{ij}\) = \(b_{ji}\) for all i and j or 1 ≤ i ≤ n, and 1 ≤ j ≤ n. Here, n is all the natural numbers. If we put i = j, then \(b_{ii}\) = 0 for all i. This means that all elements that are present diagonally in a skewsymmetric matrix are zero.
Example:
Let’s take an example of a matrix B,
\(
\begin{array}{l}
B = \left[\begin{array}{cc}
0 & 3 \\
3 & 0
\end{array}\right] \\
B^{T} = \left[\begin{array}{cc}
0 & 3 \\
3 & 0
\end{array}\right] \\
B = \left[\begin{array}{cc}
0 & 3 \\
3 & 0
\end{array}\right] \\
B = \left[\begin{array}{cc}
0 & 3 \\
3 & 0
\end{array}\right]
\end{array}\)
Here, we can see that, B^{T} = B, \(b_{12}\) = \(b_{21}\), and \(b_{11}\) = \(b_{22}\) = 0. Thus, B is a skew symmetric matrix.
Properties of Symmetric and Skew Symmetric Matrix
A matrix can be symmetric or skew symmetric only if it's a square matrix i.e, they should have the same number of rows and columns. Here are some of the important properties of symmetric and skewsymmetric matrix.
 When two skewsymmetric matrices are added, then the resultant matrix will always be a skewsymmetric matrix. Consider two skew symmetric matrices A and B such that A^{T} =A, and B^{T}=B, then we have (A + B)^{T} = (A + B)
 Since the elements that are present on the diagonal of a skewsymmetric matrix are zero, its trace also equals zero i.e., the sum of all the elements in the main diagonal is also equal to zero
 When an identity matrix of the same order is added to a skew symmetric matrix, then the resultant matrix is a matrix that is its own inverse. Consider a skew symmetric matrix B and an identity matrix I then, I + B = (I +B)^{1}
 When a scalar or a real number is multiplied with a skewsymmetric matrix, the resultant matrix will also be a skewsymmetric matrix. Consider a scalar value k, B is a skewsymmetric matrix, then the resultant matrix is also a skew symmetric matrix. ( kB)^{T} = kB.
Symmetric and Skew Symmetric Matrix Theorems
There are two important theorems related to symmetric and skew symmetric matrices. In this section, let's learn about these theorems along with their proofs.
Theorem 1: For any square matrix B with real number elements, B + B^{T} is a symmetric matrix, and B − B^{T} is a skewsymmetric matrix.
Proof:
Let A = B + B^{T}.
Taking a Transpose, A^{T} = ( B + B^{T} )^{T} = B^{T} + ( B^{T} )^{T} = B^{T} + B = B + B^{T} = A
This implies B + B^{T} is a symmetric matrix.
Next, we let C = B − B^{T}
C^{T} = ( B + ( − B^{T} ))^{T} = B^{T} + ( − B^{T} )^{T} = B^{T} − ( B^{T} )^{T} = B^{T} − B = − ( B − B^{T} ) = − C
This implies B − B^{T} is a skewsymmetric matrix.
Theorem 2: Any square matrix can be expressed as the sum of a skew symmetric matrix and a symmetric matrix. To find the sum of a symmetric and skew symmetric matrix, we use this formula:
Let B be a square matrix. then,
B = (1/2) x (B + B^{T}) + (1/2 ) x (B  B^{T}). Here, B^{T} is the transpose of the square matrix B.
 If B + B^{T} is a symmetric matrix, then (1/2) x (B + B^{T}) is also a symmetric matrix
 If B  B^{T} is a skew symmetric matrix, then (1/2 ) x (B  B^{T}) is also a skew symmetric matrix
Thus, any square matrix can be expressed as the sum of a skew symmetric matrix and a symmetric matrix
Determinant of Skew Symmetric Matrix
The determinant of a skewsymmetric matrix having an order equal to an odd number is equal to zero. So, if we see any skewsymmetric matrix whose order is odd, then we can directly write its determinant equal to 0.
Let's verify this property using a 3×3 matrix as follows:
\(B = \left[\begin{array}{ccc}
0 & a & b \\
a & 0 & m \\
b & m & 0
\end{array}\right] \)
= a (cofactor of \(b_{12}\)) + b (cofactor of \(b_{13}\))
= a ((1)^{1+2} (0)bm)) + b ((1)^{1+3} (am))
= a(1)^{3}(bm) + b(1)^{4}(am)
= a(1)(bm) + b(1)(am)
= abm + abm
= 0
Therefore, we can conclude that the determinant of a skew symmetric matrix whose order is odd, will always be zero.
Difference Between Symmetric and Skew Symmetric Matrix
There is one major difference between symmetric matrix and skew symmetric matrix. The difference has been explained in the belowgiven table:
Symmetric Matrix  Skew Symmetric Matrix 

A square matrix B which is of size n x n, is considered to be symmetric if and only if B^{T} = B  A square matrix B which is of size n x n, is considered to be symmetric if and only if B^{T} = B 
Here, \(b_{ij}\) = \(b_{ji}\)  Here, \(b_{ij}\) =  \(b_{ji}\) 
Related Articles on Symmetric and Skew Matrix
Check out the following pages related to symmetric and skew matrix.
 Matrix Calculator
 Matrix formula
 How to Solve Matrices
 Diagonal Matrix Calculator
 Transpose Matrix Calculator
Important Notes on Symmetric and Skew Matrix
Here is a list of a few points that should be remembered while studying symmetric and skew matrix.
 A square matrix that is equal to the transposed form of itself is called a symmetric matrix
 A transposed form of a matrix that is equal to the negative of that matrix is called a skewsymmetric matrix
 If the order of a skewsymmetric matrix is odd, then its determinant is equal to zero.
Examples on Symmetric and Skew Matrix

Example 1: Verify if the given matrices are symmetric and skew matrices.
a.)
\(A=\left[\begin{array}{ll}
0 & 5\\
5 & 0
\end{array}\right]\)
b.)\(B=\left[\begin{array}{ll}
2 & 1\\
1 & 2
\end{array}\right]\)
Solution:Let us take the first example:
\(
\begin{array}{l}
A = \left[\begin{array}{cc}
0 & 5 \\
5 & 0
\end{array}\right] \\
A^{T} = \left[\begin{array}{cc}
0 & 5 \\
5 & 0
\end{array}\right] \\
A = \left[\begin{array}{cc}
0 & 5 \\
5 & 0
\end{array}\right] \\
A = \left[\begin{array}{cc}
0 & 5 \\
5 & 0
\end{array}\right]
\end{array}\)Here, we can see that, A^{T} = A, \(a_{12}\) = \(a_{21}\), and \(a_{11}\) = \(a_{22}\) = 0. Thus, A is a skew symmetric matrix.
Let us take the second example:
\(
\begin{array}{l}
B = \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right] \\
B^{T} = \left[\begin{array}{cc}
2 & 1 \\
1 & 2
\end{array}\right]
\end{array}\)Here, we can see that, B^{T} = B, \(b_{12}\) = \(b_{21}\) = 1. Thus, B is a symmetric matrix.

Example 2: If
\(A=\left[\begin{array}{ll}
0 & a\\
a & 0
\end{array}\right]\)then, A is a) A skew symmetric matrix b.) A Symmetric matrix c.) None of the above d.) Symmetric and skew symmetric matrix
Solution:
\(A=\left[\begin{array}{ll}
0 & a\\
a & 0
\end{array}\right]\)A^{T} = \(\left[\begin{array}{ll}
0 & a\\
a & 0
\end{array}\right]\)A=\(\left[\begin{array}{ll}
0 & a\\
a & 0
\end{array}\right]\)After taking the transpose and the inverse of A, we can conclude that A is a skew symmetric matrix since, A^{T} = A, and \(a_{12}\) = \(a_{21}\), and \(a_{11}\) = \(a_{22}\) = 0.
Thus, option a) is the correct answer.
FAQs on Symmetric and Skew Matrix
What Are Symmetric and Skew Symmetric Matrices?
A square matrix that is equal to the transpose of that matrix is called a symmetric matrix. This is an example of a symmetric matrix:
\(A=\left[\begin{array}{ll}
2 & 7\\
7 & 8
\end{array}\right]\)
A transposed form of a matrix that is equal to the negative of that matrix is called a skewsymmetric matrix. This is an example of a skewsymmetric matrix:
\(B=\left[\begin{array}{ll}
0 & 2\\
2 & 0
\end{array}\right]\)
What Are the Properties of Symmetric and Skew Symmetric Matrices?
These are the important properties of symmetric and skew symmetric matrices:
 When two skewsymmetric matrices are added, then the resultant matrix will always be a skewsymmetric matrix.
 Since the elements that are present on the diagonal of a skewsymmetric matrix are zero, its trace also equals zero i.e., the sum of all the elements in the main diagonal is also equal to zero
 When one identity matrix is added to a skew symmetric matrix, then the resultant matrix is a matrix that is its own inverse i.e., it's invertible.
 When a scalar or a real number is multiplied with a skew symmetric matrix, the resultant matrix will also be a skewsymmetric matrix.
How Do You Know if a Matrix Is Both Symmetric and Skew Symmetric?
A null matrix or a zero matrix is the only matrix that is both symmetric and skew symmetric. Any matrix whose rows and columns are all zeroes can be considered as both symmetric and skew symmetric.
Give an example of a Matrix Which Is Both Symmetric and Skew Symmetric Matrix.
Consider the given matrix:
\(A=\left[\begin{array}{ll}
0 & 0\\
0 & 0
\end{array}\right]\)
The matrix given above is a null matrix and this is the only matrix that is both symmetric and skew symmetric.
What Is the Sum of a Symmetric and Skew Symmetric Matrix?
As per the properties of the symmetric and skew symmetric matrices, the sum of any symmetric and a skew symmetric matrix is always a square matrix. If B is a square matrix then, B = (1/2) x (B + B^{T}) + (1/2 ) x (B  B^{T}). Here, B^{T} is the transpose of the square matrix B, B + B^{T} is a symmetric matrix, and B − B^{T} is a skewsymmetric matrix.
What Is the Sum of Two Skew Symmetric Matrices?
As per the properties of the skew symmetric matrices, the sum of any two skew symmetric matrices will always be a skew symmetric matrix. Consider two skew symmetric matrices A and B, then A + B = C, C will also be a skew symmetric matrix.
What Is the Difference Between Symmetric and Skew Symmetric Matrices?
The difference between symmetric and skewsymmetric matrix has been explained in the belowgiven table:
Symmetric Matrix  Skew Symmetric Matrix 
A square matrix B which is of size n x n, is considered to be symmetric if and only if B^{T} = B  A square matrix B which is of size n x n, is considered to be symmetric if and only if B^{T} = B 
Here, \(b_{ij}\) = \(b_{ji}\)  Here, \(b_{ij}\) =  \(b_{ji}\) 
How Do You Find Symmetric and Skew Symmetric Matrix?
These are the steps to find symmetric and skew symmetric matrix:
 Step 1: Firstly, check if it's a square matrix, as only square matrices can be considered as symmetric or skew symmetric matrices.
 Step 2: Find the transpose of the given matrix.
 Step 3: Then find the negative of the given matrix.
 Step 4: If the transpose of the matrix is equal to the matrix itself, then it is a symmetric matrix.
 Step 5: If the transpose of the matrix is equal to the negative of the given matrix, then it is a skew symmetric matrix.