Orthonormal Matrix Homework: Estimating |A*A'|

[tom08]

Homework Statement

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&#039;\|?

Homework Equations

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

in the rectangular matrix section.

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.

 

[HallsofIvy]

Homework Statement

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.

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

but is there an estimate for \|A \cdot A&#039;\|?

Homework Equations

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

in the rectangular matrix section.

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.

 

[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.

 

[tom08]

can someone give me a hand?