What is a symmetric matrix definition formula properties and solved examples
Transpose of a Matrix
Let A be a matrix of order m x n; then the matrix of order n x m obtained by interchanging the rows and columns of A is called Transpose of the matrix A and is denoted by A’ or AT. For example, if
A = [fig 6] then A’= [fig 7]
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Note that A is a matrix of order 3x2 and its transpose A’ is a matrix of order 2x3.
Symmetric Matrix
A square matrix that is equal to its transpose is called a symmetric matrix. For example, a square matrix A = [aij] is symmetric if and only if aij= aji for all values of i and j, that is, if a12 = a21, a23 = a32, etc. Note that if A is a symmetric matrix then A’ = A where A’ is a transpose matrix of A.
Thus, A= [fig 4] is a symmetric matrix of order 3.
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Note that the transpose of A = A’ = [fig5] = A.
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How to Know If a Matrix is Symmetric
To know if the given matrix is symmetric or not, check the following conditions:
It should be a square matrix.
After transposing the matrix, it remains the same as that of the original matrix.
Symmetric Matrix Properties
The addition or subtraction of any two symmetric matrices will also be symmetric in nature.
The product of two symmetric matrices [A and B] doesn’t always give a symmetric matrix [AB]. The result of the product is symmetric only if two individual matrices commute (AB=BA).
The power on the symmetric matrix will also result in a symmetric matrix if the power n is integers.
The inverse of a symmetric matrix is also asymmetric.
Difference Between Symmetric and Skew-Symmetric Matrix
Determinant of Matrix
A fixed number that defines a square matrix is called the determinant of a matrix.
The process of finding the determinant of a symmetric matrix and the determinant of skew-symmetric is the same as that of a square matrix.
Matrix Inverse of a Symmetric Matrix
If A and B are two square matrices of the same order such that AB = BA = I, where I is the unit matrix of the same order as A. or B, then either B is called the inverse of A or A is called the inverse of B. The inverse of matrix A is denoted by A-1.
The inverse of a square matrix A exists if |A| is not equal to 0.
If A is nonsingular then, A-1 = \[\frac{adj(A)}{|A|}\]
Let A and B are two nonsingular Matrices then,
i) A-1. A = A. A-1 = I
ii) (A-1)-1=A
iii) (A-1)T = (AT)-1
iv) (AB)-1= B-1A-1
FAQs on Symmetric Matrix Explained with Properties and Examples
1. What is a symmetric matrix?
A symmetric matrix is a square matrix that is equal to its transpose, meaning A = AT. This implies that the elements satisfy aij = aji for all i and j.
For example:
If A =
[[1, 2],
[2, 3]],
then AT = [[1, 2], [2, 3]], so A = AT, and A is symmetric. Only square matrices can be symmetric.
2. How do you check if a matrix is symmetric?
To check if a matrix is symmetric, verify that A = AT by comparing corresponding elements across the main diagonal.
Steps:
- Ensure the matrix is square (same number of rows and columns).
- Find its transpose AT by interchanging rows and columns.
- Check whether aij = aji for all entries.
3. What is the formula for a symmetric matrix?
The defining condition (formula) for a symmetric matrix is aij = aji for all i and j, or equivalently A = AT.
For a 2 × 2 matrix:
A = [[a, b],
[c, d]]
For A to be symmetric, b = c, so the general form becomes:
[[a, b],
[b, d]].
4. Can you give an example of a symmetric matrix?
An example of a symmetric matrix is any square matrix where elements mirror across the main diagonal.
Example (3 × 3 matrix):
A = [[2, 1, 4],
[1, 3, 5],
[4, 5, 6]]
Here, a12 = a21 = 1, a13 = a31 = 4, and a23 = a32 = 5, so A = AT and the matrix is symmetric.
5. What are the properties of a symmetric matrix?
A symmetric matrix has several important algebraic properties, especially in linear algebra and eigenvalue theory.
Key properties:
- A = AT
- All eigenvalues are real
- Eigenvectors corresponding to distinct eigenvalues are orthogonal
- If A and B are symmetric, then A + B is symmetric
- kA is symmetric for any scalar k
6. What is the difference between a symmetric and skew-symmetric matrix?
The difference is that a symmetric matrix satisfies A = AT, while a skew-symmetric matrix satisfies A = −AT.
Key differences:
- Symmetric: aij = aji
- Skew-symmetric: aij = −aji
- Diagonal entries of a skew-symmetric matrix are 0
- Symmetric matrices can have any real diagonal entries
7. Are all diagonal matrices symmetric?
Yes, every diagonal matrix is a symmetric matrix because all off-diagonal elements are zero, so A = AT automatically.
For example:
A = [[3, 0],
[0, 5]]
Since all non-diagonal entries are 0, the transpose does not change the matrix, making it symmetric.
8. What are the eigenvalues of a symmetric matrix?
The eigenvalues of a real symmetric matrix are always real numbers.
Additional facts:
- Eigenvectors corresponding to different eigenvalues are orthogonal.
- A real symmetric matrix can be diagonalized using an orthogonal matrix.
- This result is known as the Spectral Theorem.
9. Can a non-square matrix be symmetric?
No, a non-square matrix cannot be symmetric because symmetry requires A = AT, which is only possible for square matrices.
If a matrix has different numbers of rows and columns, its transpose will have different dimensions, so equality is impossible. Therefore, symmetry is defined only for square matrices.
10. How do you form a symmetric matrix from any square matrix?
A symmetric matrix can be formed from any square matrix A using the formula (A + AT)/2.
Steps:
- Find the transpose AT.
- Add A and AT.
- Divide the result by 2.
