Are the Row Vectors of a Matrix with Orthonormal Columns Also Orthonormal?

[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

 

[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.

 

[HallsofIvy]

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.