Question

Prove transpose of orthogonal matrix is orthogonal

Answer 1

Answer:

orthogonal matrix is a real square matrix whose columns and rows are orthogonal unit vectors (orthonormal vectors). is the identity matrix.

Answer 2

Given,

• AA' = A'A = I

To find,

• (A')'A' = I

Solution,

This is proved that the transpose of the orthogonal matrix is orthogonal.

Let A be any square matrix of order n,

It is given that A is an orthogonal matrix, then

                  AA' = A'A = I  

Taking transpose both sides, we get

                  (AA')' = (A'A)' = (I)'   [ I' = I]

                   (A')' (A)' = I             [$(AB)' = B'A'$]

                         AA' = I              [$(A')' = A$]

Hence proved that the transpose of the orthogonal matrix is orthogonal.