A matrix is called a symmetric matrix if its transpose is equal to the matrix itself. Only a square matrix is symmetric because in linear algebra equal matrices have equal dimensions.
How do you know if a matrix is symmetric? Generally, the symmetric matrix is defined as
A = AT
Where A is any matrix, and AT is its transpose.
If aij denotes the entries in an i-th row and j-th column, then the symmetric matrix is represented as
aij = aji
Where all the entries of a symmetric matrix are symmetric with respect to the main diagonal. The symmetric matrix examples are given below:
2 x 2 square matrix :
3 x 3 square matrix :
What is the Transpose of a Matrix?
A matrix “M” is said to be the transpose of a matrix if the rows and columns of a matrix are interchanged. In this case, the first row becomes the first column, and the second row becomes the second column and so on. The transpose of a matrix is given as “MT “.
Consider the above matrix A and B
Then, the transpose of a matrix is given by
When you observe the above matrices, the matrix is equal to its transpose.
Therefore, the symmetric matrix is written as
A = AT and B = BT
Symmetric Matrix Inverse
Since the symmetric matrix is taken as A, the inverse symmetric matrix is written as A-1, such that it becomes
A × A-1 = I
Where “I” is the identity matrix.
If a matrix contains the inverse, then it is known as invertible matrix, and if the inverse of a matrix does not exist, then it is called a non-invertible matrix.
The symmetric matrix inverse can be found using two methods. They are
- Adjoint Method
- Gauss-Jordan Elimination method.
It is noted that inverse of the given symmetric matrix is also a symmetric matrix.
Symmetric Matrix Determinant
Finding the determinant of a symmetric matrix is similar to find the determinant of the square matrix. A determinant is a real number or a scalar value associated with every square matrix. Let A be the symmetric matrix, and the determinant is denoted as “det A” or |A|. Here, it refers to the determinant of the matrix A. After some linear transformations specified by the matrix, the determinant of the symmetric matrix is determined.
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Properties of Symmetric Matrix
Symmetric matrix is used in many applications because of its properties. Some of the symmetric matrix properties are given below :
- The symmetric matrix should be a square matrix.
- The eigenvalue of the symmetric matrix should be a real number.
- If the matrix is invertible, then the inverse matrix is a symmetric matrix.
- The matrix inverse is equal to the inverse of a transpose matrix.
- If A and B be a symmetric matrix which is of equal size, then the summation (A+B) and subtraction(A-B) of the symmetric matrix is also a symmetric matrix.
- A scalar multiple of a symmetric matrix is also a symmetric matrix.
- If the symmetric matrix has distinct eigenvalues, then the matrix can be transformed into a diagonal matrix. In other words, it is always diagonalizable.
- For every distinct eigenvalue, eigenvectors are orthogonal.
Symmetric and Skew Symmetric Matrix
A matrix is Symmetric Matrix if transpose of a matrix is matrix itself.
Consider a matrix A, then
Transpose of A = A
A matrix is Skew Symmetric Matrix if transpose of a matrix is negative of itself.
Consider a matrix A, then
Transpose of A = – A
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Show that the product ATA is always a symmetric matrix.
Consider a matrix,
Now take the transpose of a matrix A,
To prove : The product of ATA is always a symmetric matrix.
So, taking the transpose of ATA ,
The transpose of ATA is a symmetric matrix.
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