Definition Formula Properties and Solved Examples of Symmetric and Skew Symmetric Matrices
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 in Linear Algebra
1. What is a symmetric matrix?
A symmetric matrix is a square matrix that is equal to its transpose, that is, A = AT. This means the elements across the main diagonal are equal.
- Condition: aij = aji
- It must be a square matrix.
- Example: If A = [[1, 2], [2, 3]], then AT = A, so it is symmetric.
2. What is a skew symmetric matrix?
A skew symmetric matrix is a square matrix that satisfies AT = −A. This means each element is the negative of its transpose counterpart.
- Condition: aij = −aji
- All diagonal elements are 0.
- Example: A = [[0, 2], [−2, 0]] is skew symmetric.
3. What is the difference between symmetric and skew symmetric matrix?
The main difference is that a symmetric matrix satisfies A = AT, while a skew symmetric matrix satisfies AT = −A.
- Symmetric: aij = aji
- Skew symmetric: aij = −aji
- Diagonal elements: any value in symmetric, but 0 in skew symmetric.
4. How do you check if a matrix is symmetric?
To check if a matrix is symmetric, verify whether A = AT.
- Step 1: Ensure the matrix is square.
- Step 2: Find the transpose AT.
- Step 3: Compare corresponding elements.
- If all aij = aji, the matrix is symmetric.
5. How do you check if a matrix is skew symmetric?
A matrix is skew symmetric if it satisfies AT = −A and all diagonal elements are 0.
- Step 1: Confirm the matrix is square.
- Step 2: Find AT.
- Step 3: Check whether AT equals −A.
- Step 4: Verify diagonal entries are zero.
6. Why are diagonal elements zero in a skew symmetric matrix?
Diagonal elements of a skew symmetric matrix are zero because the condition aii = −aii implies aii = 0.
- From AT = −A
- For diagonal entries: aii = −aii
- So 2aii = 0 ⇒ aii = 0
7. Can a matrix be both symmetric and skew symmetric?
A matrix can be both symmetric and skew symmetric only if it is the zero matrix.
- If A = AT and AT = −A
- Then A = −A
- So 2A = 0 ⇒ A = 0
8. What is an example of a symmetric and skew symmetric matrix?
An example of a symmetric matrix is [[2, 5], [5, 1]], and an example of a skew symmetric matrix is [[0, 3], [−3, 0]].
- Symmetric: Off-diagonal elements are equal.
- Skew symmetric: Off-diagonal elements are negatives and diagonals are zero.
9. What is the formula to decompose a matrix into symmetric and skew symmetric parts?
Any square matrix A can be written as the sum of a symmetric and a skew symmetric matrix using A = (A + AT)/2 + (A − AT)/2.
- Symmetric part: (A + AT)/2
- Skew symmetric part: (A − AT)/2
- This decomposition works for every square matrix.
10. Are eigenvalues of a symmetric matrix always real?
Yes, the eigenvalues of a real symmetric matrix are always real numbers.
- This is a key result in linear algebra.
- Real symmetric matrices also have orthogonal eigenvectors.
- This property is widely used in matrix theory and applications.
