(1) If A is an n x n matrix, then prove that (A^T)A (i.e., A transpose multiplied by A) is symmetric.(adsbygoogle = window.adsbygoogle || []).push({});

(2) Let S be the set of symmetric n x n matrices. Prove that S is a subspace of M, the set of all n x n matrices.

(3) What is the dimension of S?

(4) Let the function f : M-->S be defined by f(X)=(X^T)X-I. Compute Df(A).

(5) Show that Df(A) is onto when A is an orthogonal matrix.

(6) Prove that O, the set n x n orthogonal matrices, is a manifold of dimension (n^2-n)/2.

(7) Show that the tangent space to O at I is the space of skew-symmetric matrices. Recall that the skew-symmetric matrices satisfy H^T=-H.

(8) Is this the same dimension as in (6)?

I need to write an easily-readable solution for a freshman-level theoretical calculus/geometry course. Can anyone please help? Thanks.

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# Symmetric Matrices and Manifolds Answer Guide

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