Orthogonal Matrix

A matrix A is defined as orthogonal if its inverse, A^-1, is equal to its transpose, A^T.

The set of all n-dimensional orthogonal matrices is denoted by the symbol O_n.

Note. Only invertible matrices can be orthogonal, meaning orthogonal matrices form a subset of O_n within the set GL_n(R) of invertible n x n matrices.

In an orthogonal matrix, the product of matrix A with its transpose A^T equals the identity matrix I_n of order n.

Proof

The proof is simple; one of the key properties of invertible matrices is as follows:

For orthogonal matrices, the inverse matrix is identical to the transpose, so A^-1 = A^T.

This leads to the conclusion that AA^T = I_n.

A practical example

The following square matrix is an orthogonal matrix.

To confirm this, simply multiply A by its transpose, A^T.

The result of A · A^T is the identity matrix I₂.

Consequently, it follows that A^T = A^-1.

Why? This is a property of inverse matrices: the product of an invertible matrix (A) with its inverse (A^-1) results in the identity matrix I. Therefore, if AA^T = I and AA^-1 = I, then A^T = A^-1.

This verifies that it is indeed an orthogonal matrix.