Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Symmetric Matrices and Manifolds Answer Guide

  1. Nov 14, 2005 #1
    (1) If A is an n x n matrix, then prove that (A^T)A (i.e., A transpose multiplied by A) is symmetric.
    (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.
  2. jcsd
  3. Nov 14, 2005 #2
    Surely if you've been asked to provide the answers, shouldn't you be able to come up with solutions?

    Or have I missed the point here?
  4. Nov 14, 2005 #3
    Unfortunately, I was trained as an applied mathematician with few abstract or theoretical courses. This is my first year at this job and for the first time I am completely lost. Can you help?
  5. Nov 14, 2005 #4


    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    the first few of these problems follow immediateoly from the definitions of the concepts. so review those definitions.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Symmetric Matrices and Manifolds Answer Guide