Symmetric Matrix
Symmetric Matrix is known that similar matrices have similar dimensions, thus only the square matrices can either be symmetric or skew-symmetric. In other words, it can be said that both a symmetric matrix and a skew-symmetric matrix are square matrices and the difference between a symmetric matrix and a skew-symmetric matrix is that symmetric matrix is always equivalent to its transpose while the 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 = - M\[^{T}\].
When a symmetric matrix and skew-symmetric matrix are summed up, the resultant matrix is always square.
Meaning of a Symmetric Matrix
A matrix cab only is stated as a symmetric matrix if its transpose is equivalent to the matrix itself. It should be remembered that only a square matrix is a symmetric matrix because in linear algebra similar matrices have similar dimensions.
Generally, the symmetric matrix is expressed as
M = M\[^{T}\]
Where M is any matrix and M\[^{T}\] is
transpose of that matrix.
If a(i,j) represents any
elements in an ith column
and jth rows,
then the symmetric matrix is expressed as
aᵢⱼ = aⱼᵢ
Where every element of a asymmetric matrix is symmetric concerning the main diagonal whereas A square Matrix A can be defined as skew-symmetric if aij = aji for all the values of i and j. So, we can also say that the matrix P is said to be the skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A, In other words, A\[^{T}\] = −A.
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 the 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 condition is valid for each value 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.
2. 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.
FAQs on Symmetric Matrix and Skew Symmetric Matrix
1. What is the formula for the addition of asymmetric and skew-symmetric matrices?
As we have already established that the sum of a symmetric matrix and skew-symmetric matrix is always a square matrix. So, the below-mentioned formula will be used to find the sum of the symmetric matrix and skew-symmetric matrix.
Let M be a square matrix then,
M = (½) × ( M + M’) + (½) ×( M - M’)
M’ is known as the transpose of a matrix.
1/2( M + M’) is the symmetric matrix
1/2( M - M’) is the skew-symmetric matrix
2. Give the mathematical formula for a skew-symmetric matrix?
Let us say a square Matrix A is defined as skew-symmetric if aᵢⱼ = aⱼᵢ for all the values of i and j. So, 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). So, in the skew matrix symmetric if all the elements present in the main diagonal of matrices A are zero and also a₁₂ = -c and a₂₁ = -c which implies that a₁₂ = a₂₁. This condition is valid for all values of i and j.
3. What are the properties of the Skew-Symmetric matrix?
Some of the properties of a skew-symmetric matrix are:
When the two skew-matrices are added, the resultant matrix will always be a skew-matrix.
The result of the scalar product of a skew-symmetric matrix is always a skew-symmetric matrix.
The elements included in the main diagonal of the skew matrix are always equal to zero. Thus, the total of all the elements of the skew matrix in the main diagonal is equal to zero.
When both the identity matrix and the skew-symmetric matrix are added, the matrix obtained is invertible.
The determinants of the skew-symmetric matrices are always non-negative.
4. What are the properties of a skew-symmetric matrix?
There are several applications of symmetric matrices due to their properties. Some of the symmetric matrix properties are:
It is necessary for the symmetric matrix to be square.
The eigenvalue of a symmetric matrix should always be given in a real number.
If the matrix given is invertible, then the inverse matrix should be considered a symmetric matrix
The inverse matrix shall always be equivalent to the inverse of a transpose matrix.
If P and Q are two symmetric matrices of equal size, then the total of (P + Q) and subtraction of (P- Q) of the symmetric matrix shall also be the symmetric matrix.
5. What are the determinant 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 condition that Det (MT) = det (-M) = (-1)n det(M) and the inverse of a skew-symmetric matrix is not possible to calculate as the determinant of it having odd order is zero and hence it is singular. The property of the determinants of the skew-symmetric matrix can be verified using an example of a 3 by 3 matrix. The determinants of the skew matrix can be found out using co-factors and thus can state that its determinant is equivalent to zero.
