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.Originally Posted by Math Help
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).