Thread: Orthognal matrices are compact

HI,
How can I prove the set of compact matrices are compact

Do you mean Orthogonal matrices are compact You wrote:
How can I prove the set of compact matrices are compact

Yes.Set of orthogonal matrices are compact.I now why they are bounded but does not know why they are closed.

Please give us an equation to define the matrix.

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).

Can you explain how you know they are bounded? Thanks.

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.