# Homework Help: Orthonormal matrix

1. Jan 24, 2010

### tom08

1. The problem statement, all variables and given/known data

If A is a rectangular n*m matrix (n>m) , and all the columns of A is orthonormal.

I know that A'*A=I, where A' stands for its transpose.

but A*A'<>I as I've learned from wiki. but is there an estimate for $$\|A \cdot A'\|$$?

2. Relevant equations

http://en.wikipedia.org/wiki/Orthogonal_matrix

in the rectangular matrix section.

3. The attempt at a solution

I have tried to write A in component form to find any hints. but i failed to solve the problem. but when i test A*A' , i always find that norm(A*A')=1, could you help me to explain it? Thank you in advance.

2. Jan 24, 2010

### HallsofIvy

What? The Wikipedia site you link to below clearly says, "$Q^TQ= QQ^T= I$. Alternatively $Q^T= Q^{-1}$".

3. Jan 24, 2010

### tom08

no, please look at my wiki link in the last secion, "rectangular matrix"

if Q is not square, but column orthonormal. let Q be an n-by-m matrix, and (m<n),

then Q'*Q=I, but Q*Q'<>I.

so i want to find out an upper bound of ||I-Q*Q'|| w.r.t m, where m is the number of orthonormal columns of Q.

4. Jan 25, 2010

### tom08

can someone give me a hand?