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A symmetric matrix is a type of square matrix that remains unchanged when transposed. In simpler terms, a matrix A is called symmetric if it is equal to its transpose, denoted as AT. This means that the matrix satisfies the condition A = AT.
Whereas, a skew-symmetric matrix (or antisymmetric matrix) is a special type of square matrix where each element is equal to the negative of its corresponding element in the transpose matrix. For a square matrix A, its transpose is denoted by AT. A matrix A is considered skew-symmetric if it satisfies the condition A = – AT.
In this article, we will learn more about Symmetric Matrix & Skew Symmetric Matrix.
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What are the Matrices?
A Matrix is a rectangular array of m × n numbers, which may be either real or complex. These numbers are set in the form of m horizontal lines and n vertical lines. These are altogether defined as a matrix of order m by n. It is represented as m × n Matrices. The rectangular array is enclosed in either () or [] brackets. This article will discuss the types of Matrices.
What is a Symmetric Matrix?
A symmetric matrix is a type of square matrix that remains unchanged when transposed. In simpler words, a square matrix A = [aij] is called a symmetric matrix if aij = aji, for all i,j values
Example of the symmetric matrix:
|
A = |
1
3 4 |
3
2 8 |
4
8 6 |
|
AT = |
1
3 4 |
3
2 8 |
4
8 6 |
Therefore, Matrix A is an example of a symmetric matrix
Note: Matrix A is symmetric if AT = A, where AT is the transpose of the matrix.
Symmetric matrices are fundamental in various fields, especially in machine learning, data analysis, and optimisation, due to their unique properties that simplify complex computations.
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What is a Skew-Symmetric Matrix?
A skew-symmetric matrix (or antisymmetric matrix) is a special type of square matrix where each element is equal to the negative of its corresponding element in the transpose matrix. A square matrix A = [aij] is a skew-symmetric matrix if aij = -aji, for all values of i,j.
If we put i = j, then,
aii = -aii
⇒ 2 aii = 0
⇒ aii = 0
Thus, in a skew-symmetric matrix, all diagonal elements are zero.
Example of a Skew-symmetric matrix:
| 0
2 3 |
-2
0 5 |
-3
-5 0 |
is an example of skew-symmetric matrices.
Note: A square matrix A is said to be a skew-symmetric if A’ = – A.
Skew-symmetric matrices have various applications across different fields, including machine learning, physics, and statistical analysis, making them an important concept to grasp.
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Properties of Symmetric Matrices
Symmetric matrices possess unique properties that make them crucial in various mathematical and computational applications. Below are some key properties of symmetric matrices:
- Sum and Difference: The sum and difference of two symmetric matrices result in another symmetric matrix. If A and B are symmetric matrices, then both A+B and A−B are symmetric.
- Product of Symmetric Matrices: The product of two symmetric matrices A and B is said to be symmetric if and only if the matrices commute, i.e., AB = BA.
- Powers of a Symmetric Matrix: For any integer n, if A is a symmetric matrix, then An (A raised to the power n) is also symmetric.
- Eigenvalues: The eigenvalues of a symmetric matrix are always real. Moreover, for positive definite symmetric matrices, the eigenvalues are all positive.
- Determinant: The determinant of a symmetric matrix is the same as the determinant of its transpose. This highlights a fundamental symmetry in the matrix’s structure.
- Adjoint Matrix: The adjoint (or adjugate) of a symmetric matrix is also symmetric. This property is particularly useful in various mathematical transformations.
- Inverse of a Symmetric Matrix: If a symmetric matrix is invertible, then its inverse is also symmetric. This property helps maintain symmetry in systems of linear equations and other applications.
Properties of Skew-Symmetric Matrices
A matrix is considered skew-symmetric if it meets two key conditions: it must be a square matrix (having an equal number of rows and columns) and it must be equal to the negative of its transpose. Below given are important properties of skew-symmetric matrices:
- Addition of Skew-Symmetric Matrices: When two skew-symmetric matrices are added, the resulting matrix is also skew-symmetric.
- Trace Equals Zero: The trace of a skew-symmetric matrix is always zero. This means the sum of all the elements along the main diagonal is equal to zero.
- Eigenvalues of Real Skew-Symmetric Matrices: For a real skew-symmetric matrix A, the real eigenvalues are zero. The nonzero eigenvalues of a skew-symmetric matrix are purely imaginary, meaning they are not real numbers.
- Multiplication by a Scalar: Multiplying a skew-symmetric matrix by a scalar (real number) results in another skew-symmetric matrix.
- Invertibility of I+A: For any real skew-symmetric matrix A, the matrix I+A (where I is the identity matrix) is always invertible.
- Square of a Skew-Symmetric Matrix: For any real skew-symmetric matrix A, the square of the matrix, A2, is a symmetric negative semi-definite matrix. This property is particularly useful in optimization and theoretical analysis.
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Difference Between Symmetric Matrices and a Skew-symmetric Matrices
Below given are the key differences between Symmetric and Skew-symmetric matrices.
| Symmetric Matrix | Skew Symmetric Matrix |
| A square matrix B which is of size n × n, is considered to be symmetric if and only if BT = B. | A square matrix B which is of size n × n, is considered to be symmetric if and only if BT = -B. |
| Here, bij = bji. | Here, bij = – bji. |
| There is nothing specific about the determinant of a symmetric matrix. | The determinant of a skew-symmetric matrix of odd order is 0. |
| The eigenvalues of the symmetric matrix are real. | The eigenvalues of the skew-symmetric matrix are purely imaginary. |
| The elements of the principal diagonal may be any elements. | The elements of the principal diagonal are always zeros. |
Some Important Conclusions on Symmetric and Skew-Symmetric Matrices
- If A is any square matrix,
- then A + A’ is a symmetric matrix
- And, A – A’ is a skew-symmetric matrix.
- Every square matrix can be uniquely expressed as the sum of a symmetric and skew-symmetric matrix.
- If A and B are symmetric matrices, then A & B commute, i.e. AB is symmetric AB = BA.
- The matrix B’AB is symmetric or skew-symmetric in correspondence if A is symmetric or skew-symmetric.
- All positive integral powers of a symmetric matrix are symmetric.
- Positive odd integral powers of a skew-symmetric matrix are skew-symmetric, and positive even integral powers of a skew-symmetric matrix are symmetric.
Symmetric Matrix & Skew Symmetric Matrix: FAQs
What is the key difference between symmetric and skew-symmetric matrices?
A symmetric matrix is equal to its transpose (A=AT), while a skew-symmetric matrix is equal to the negative of its transpose (A=−AT). Symmetric matrices have real, often positive eigenvalues, whereas skew-symmetric matrices have purely imaginary or zero eigenvalues.
Are all symmetric matrices invertible?
Not all symmetric matrices are invertible. A symmetric matrix is invertible only if its determinant is non-zero. If the matrix is positive definite, it is guaranteed to be invertible.
Can a skew-symmetric matrix have non-zero diagonal elements?
No, a skew-symmetric matrix cannot have non-zero diagonal elements because the diagonal elements must be equal to their own negatives. Therefore, all diagonal elements in a skew-symmetric matrix are zero.
