Symmetric Matrix

What is Symmetric Matrix?
A square matrix is said to be a symmetric matrix when the matrix and its transpose are equal.
Symbolically, it can be represented as A = AT.  Transpose means that the rows are written as columns and columns are written as rows.

Inverse of a Symmetric Matrix
Let us see how inverse of a symmetric matrix is calculated in two special cases namely diagonal matrix and 2x2 matrixes.

Inverse of a Diagonal Matrix
Diagonal matrix is one of the special cases of symmetric matrix. The matrix elements present in positions other than the main diagonal will be zero in the diagonal matrix.  By replacing every element in the main diagonal of the diagonal matrix with its corresponding reciprocal, the inverse of a diagonal symmetric matrix is obtained.

The equation XX-1  =  X-1X = I confirms that X-1  is the inverse of X.  In this expression, I represent identity matrix.

Your professor might ask you to find inverse of a diagonal matrix with one of the elements in the main diagonal as zero. Be careful, it’s a tricky question. If any of the diagonal elements is zero, then the inverse of that diagonal symmetric matrix cannot be calculated i.e., it has no inverse.

Then the determinant, denoted by |X| is calculated by the formula
|X| = X11X22 – X12X21

Then, the inverse of symmetric matrix is given by

Determinant of Symmetric Matrix
The determinant of symmetric matrix is n (n+1)/2 scalars, which represents all the entries in the matrix that are above the main diagonal and that are present on the main diagonal.

Properties of Symmetric Matrices

Symmetric Matrix properties are:

• If X and Y are two symmetric matrices of size m x m, then
o X+Y is also a symmetric matrix as (X+Y)T = XT+YT = (X+Y).
o XY is not symmetric as (XY)T = YTXT = YX which is not equal to XY.

• If the given matrix is a diagonal matrix, then it is a symmetric matrix.

• The product of a symmetric matrix and its transpose is a symmetric matrix.