Home | Math | What is an Orthogonal Matrix?-Example of Orthogonal Matrix

# What is an Orthogonal Matrix?-Example of Orthogonal Matrix

July 28, 2022 written by Azhar Ejaz

A square matrix ‘A’ over R is called an orthogonal matrix

If AAt=I

OR

At=A-1

A = a square matrix

At=transpose of A

I=identity matrix of the same order as ‘A’

A-1=inverse of A

The product of the square matrix and it is transposed gives an identity matrix same order is called an orthogonal matrix. Or we can say A square matrix with real numbers is called an orthogonal matrix if its transpose is equal to its inverse of a matrix.

For example:-

Solution:

Given matrix

According to the definition of an orthogonal matrix

Prove that

We know that

Cos2 α+ sin2α=1

-sinα cosα+sinα cosα=0

Summary

• Every identity matrix is an orthogonal matrix
• If A is an orthogonal matrix then A-1 is also an orthogonal matrix
• If A is an orthogonal matrix then At is orthogonal
• If A is an orthogonal matrix then An is orthogonal
• The product of two orthogonal matrices is also orthogonal
File Under: