# Orthognal matrices are compact

• Apr 27th 2005, 11:29 AM
akscola
Orthognal matrices are compact
HI,
How can I prove the set of compact matrices are compact
• Apr 27th 2005, 12:21 PM
Math Help
Do you mean Orthogonal matrices are compact You wrote:
How can I prove the set of compact matrices are compact
• Apr 28th 2005, 03:33 AM
akscola
Orthognal matrices are compact
Yes.Set of orthogonal matrices are compact.I now why they are bounded but does not know why they are closed.
• Apr 29th 2005, 08:30 AM
Shmuel
Please give us an equation to define the matrix.
• Apr 29th 2005, 09:28 AM
hpe
Quote:

Originally Posted by Math Help
Do you mean Orthogonal matrices are compact You wrote:
How can I prove the set of compact matrices are compact

Let O = set of orthogonal nxn matrices, that is O = {A | A'A = E}, where A' = transpose of A and E = unit matrix.

The condition A'A = E actually is a set of n^2 equations that have to hold. The left hand side of each equation is a continuous function on the set of all nxn matrices (just a quadratic form, actually). The right hand side of each equation is either 0 or 1, a closed set.

Preimages of closed sets under continuous functions are closed.

Finite intersections of closed sets are closed.

Done: O is closed. (You already proved it's bounded).
• May 23rd 2009, 03:27 PM
deannalouise
Can you explain how you know they are bounded? Thanks.
• May 24th 2009, 04:50 PM
hpe
Quote:

Originally Posted by deannalouise
Can you explain how you know they are bounded? Thanks.

Trace(O'O) = n. That's the sum of squares of the entries of the matrix O.