Key Properties and Differences of Symmetric & 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 vs Skew Symmetric Matrix Explained
1. What is a symmetric matrix? Please provide an example.
A symmetric matrix is a square matrix that is equal to its own transpose. This means that if we have a matrix A, it is symmetric if A = Aᵀ. In a symmetric matrix, the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column (aᵢⱼ = aⱼᵢ). The elements are symmetric with respect to the main diagonal.
For example, the following 3x3 matrix is symmetric:
A =
| 1 7 3 |
| 7 4 -5 |
| 3 -5 6 |
2. What is a skew-symmetric matrix and how is it identified?
A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. For a matrix A to be skew-symmetric, it must satisfy the condition A = -Aᵀ. This implies that the element in the i-th row and j-th column is the negative of the element in the j-th row and i-th column (aᵢⱼ = -aⱼᵢ). A key identifying feature is that all elements on the main diagonal of a skew-symmetric matrix are always zero.
3. What is the primary difference between a symmetric and a skew-symmetric matrix?
The primary difference lies in their relationship with their transpose:
- A symmetric matrix is equal to its transpose (A = Aᵀ). Its elements are mirrored across the main diagonal (aᵢⱼ = aⱼᵢ).
- A skew-symmetric matrix is equal to the negative of its transpose (A = -Aᵀ). Its off-diagonal elements are negatives of each other (aᵢⱼ = -aⱼᵢ), and its main diagonal elements must be zero.
4. Why must the diagonal elements of any skew-symmetric matrix be zero?
The diagonal elements of a skew-symmetric matrix must be zero due to the fundamental property that defines it: aᵢⱼ = -aⱼᵢ. For any element on the main diagonal, the row index 'i' is equal to the column index 'j'. So, for a diagonal element aᵢᵢ, the condition becomes aᵢᵢ = -aᵢᵢ. If we rearrange this equation, we get 2aᵢᵢ = 0, which means aᵢᵢ must be 0. This holds true for all elements along the main diagonal.
5. What are some key properties of a symmetric matrix?
Here are some key properties of a symmetric matrix:
- It must be a square matrix.
- The sum or difference of two symmetric matrices of the same size is also a symmetric matrix.
- If a symmetric matrix is invertible, its inverse is also a symmetric matrix.
- All eigenvalues of a real symmetric matrix are real numbers.
6. How can any square matrix be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix?
According to a fundamental theorem in matrix algebra, any square matrix 'A' can be uniquely expressed as the sum of a symmetric matrix (P) and a skew-symmetric matrix (Q). The formulas for P and Q are:
- Symmetric part (P) = ½(A + Aᵀ)
- Skew-symmetric part (Q) = ½(A - Aᵀ)
Thus, A = P + Q = ½(A + Aᵀ) + ½(A - Aᵀ). This decomposition is unique for any given square matrix.
7. If a matrix is both symmetric and skew-symmetric, what kind of matrix must it be?
If a matrix 'A' is both symmetric and skew-symmetric, it must be a zero matrix (or null matrix). This is because the conditions for both must be met simultaneously:
- Symmetric: A = Aᵀ
- Skew-symmetric: A = -Aᵀ
By substituting the first equation into the second, we get A = -A. This simplifies to 2A = 0, which implies that every element of matrix A must be zero.
8. What is the significance of the determinant of a skew-symmetric matrix of odd order?
The determinant of any skew-symmetric matrix with an odd order (e.g., a 3x3, 5x5, etc.) is always zero. This property is significant because a determinant of zero implies that the matrix is singular, meaning it does not have a multiplicative inverse. For even-ordered skew-symmetric matrices, the determinant is a non-negative value and can be non-zero.
9. Can a rectangular (non-square) matrix be classified as symmetric or skew-symmetric?
No, a rectangular matrix cannot be classified as symmetric or skew-symmetric. These concepts are defined exclusively for square matrices. The reason is that the definition requires a comparison between the matrix A and its transpose Aᵀ. If A is an m x n matrix where m ≠ n, its transpose Aᵀ will be an n x m matrix. Since A and Aᵀ have different dimensions, they cannot be equal or be the negative of each other.
10. Is an 'antisymmetric matrix' the same as a 'skew-symmetric matrix'?
Yes, the terms antisymmetric matrix and skew-symmetric matrix are used interchangeably in linear algebra. Both names refer to a square matrix whose transpose is equal to its negative, fulfilling the condition A = -Aᵀ.
