what is a symmetric matrix?

A symmetric matrix is a square matrix that remains unchanged when its rows and columns are interchanged. In other words, a matrix is symmetric if it is equal to its transpose.

Condition for a Symmetric Matrix:

A = Aᵀ

This means:

aij = aji

for all values of i and j.

Example:

A =1 2 3

      2 4 5

      3 5 6

This matrix is symmetric because:

Element in Row 1, Column 2 = 2 and Element in Row 2, Column 1 = 2

Element in Row 1, Column 3 = 3 and Element in Row 3, Column 1 = 3

Element in Row 2, Column 3 = 5 and Element in Row 3, Column 2 = 5

The elements on opposite sides of the main diagonal are equal.

Key Features:

It must be a square matrix.

Its transpose is equal to the original matrix.

Corresponding elements satisfy aij = aji.

In simple terms: A symmetric matrix is a square matrix in which the elements above and below the main diagonal are identical.