Orthonormal row vectors

1. Mar 19, 2006

mathboy20

Hi

Given A an n x n matrix where the columns are the set of standard vectors S = {e1, e2,.....e_n} in R^n

These vectors are orthonormal according to the definition since

<e_i, e_j> = 0, i \neq j

<e_i,e_i> = 1

Since the columns of A then they are orthonormal according to the definition, and therefore the matrix A is orthogonal matrix which implies that

A^T A = I <---> A^ (-1) = A^T, and so on.

I need to show that the row vectors of of A are orthonormal too.

Any idears on how?

My own idear is that

Since

A^ (-1) = A^T, then the columns in invertible A become the rows in transposed A, and the these orthonormal according to the definition?

Sincerely Yours
Mathboy20

2. Mar 19, 2006

0rthodontist

I don't know what your reasoning is, but with A^(-1) = A^T then you also know that (A^T)^(-1) = A. A^T*A = A*A^T. It follows from there.

3. Mar 20, 2006

HallsofIvy

Staff Emeritus
Actually, the way you stated the problem, its trivial. If the columns of the matrix are just the standard basis vectors e1, etc. then the rows are those basis vectors too! More likely you want to prove that if columns are any set of orthonormal vectors, then the rows are too. Orthodontist's hint is good.