Symmetric Matrix – Infinity Learn

Symmetric and Skew Symmetric Matrix

Symmetric Matrix – Infinity Learn: A symmetric matrix is one in which the elements on the diagonal are the same as the elements below the diagonal and above the diagonal. A skew symmetric matrix is one in which the elements on the diagonal are the opposite of the elements below the diagonal and above the diagonal.

Transpose of a Matrix

Transposing a matrix is the process of exchanging the rows and columns of a matrix. For example, the transpose of the matrix

[1 2 3]

would be

[3 2 1]

Symmetric Matrix

A symmetric matrix is a square matrix in which all the entries of the corresponding row and column are equal.

How to Know If a Matrix is Symmetric

A matrix is symmetric if and only if its transpose is equal to its inverse.

Symmetric Matrix Properties

• A symmetric matrix is a square matrix in which the entries on the main diagonal (upper left to lower right) are all equal, and the entries off the main diagonal are all equal in magnitude but opposite in sign.
• A symmetric matrix has the following properties:
• The matrix is symmetric about the main diagonal.
• The matrix is positive definite if all the off-diagonal entries are positive, and negative definite if all the off-diagonal entries are negative.
• The matrix is Hermitian if it is real and symmetric.
• The matrix is orthogonal if it is real and symmetric, and its columns are orthonormal.

Determinant of Matrix

The determinant of a 2×2 matrix is the product of the diagonal elements minus the product of the off-diagonal elements.

A = |a| |b|

det A = a*b – b*a

Matrix Inverse of a Symmetric Matrix

A matrix is symmetric if and only if its transpose is the same as its inverse.

A matrix inverse is a mathematical operation that finds the inverse of a matrix. Given a square matrix A, the inverse of A, denoted A−1, is a matrix such that AA−1 = I, where I is the identity matrix. The inverse of a matrix is unique if the matrix is invertible.

If A is a symmetric matrix, then its inverse is also symmetric. To find the inverse of a symmetric matrix, we use the following algorithm:

1. Find the determinant of A.

2. If the determinant is zero, the matrix is not invertible and the algorithm stops.

3. If the determinant is not zero, find a matrix B such that AB is the identity matrix.

4. The inverse of A is then A−1 = B−1AB.

The following examples demonstrate how to find the inverse of a symmetric matrix.

Example 1: Find the inverse of the matrix

We first find the determinant of A.

The determinant of A is not zero, so we continue to step 3.

We find a matrix B such that AB is the identity matrix.

The inverse of A is then A−1 = B−1AB.

Example 2: Find the inverse of the matrix

We first find the determinant of A.

The determinant of A is zero, so the matrix is not invertible and the algorithm stops.