Symmetric Matrices and Manifolds Answer Guide

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Discussion Overview

The discussion revolves around a series of mathematical problems related to symmetric matrices, orthogonal matrices, and their properties within the context of linear algebra and manifold theory. The scope includes theoretical proofs, definitions, and applications relevant to a freshman-level calculus or geometry course.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Post 1 outlines several problems, including proving that (A^T)A is symmetric, showing that the set of symmetric matrices is a subspace, and determining the dimension of this set.
  • Post 1 also asks for the computation of the derivative of a function defined on matrices and explores properties of orthogonal matrices and their tangent spaces.
  • Post 2 questions the need for assistance in solving the problems, suggesting that the poster should be able to derive solutions independently.
  • Post 3 expresses a lack of familiarity with abstract concepts due to a background in applied mathematics, indicating a need for help with the theoretical aspects of the problems.
  • Post 4 suggests that the initial problems can be resolved by reviewing the definitions of the relevant concepts.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the necessity of assistance, with some expressing frustration and others indicating a genuine need for help. The discussion remains unresolved regarding the best approach to tackle the problems presented.

Contextual Notes

Some participants may have differing levels of familiarity with the theoretical aspects of linear algebra and manifold theory, which could affect their ability to engage with the problems effectively.

Who May Find This Useful

This discussion may be useful for students in introductory courses on linear algebra, calculus, or geometry, particularly those struggling with theoretical concepts related to matrices and manifolds.

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(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.
 
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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?
 
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?
 
the first few of these problems follow immediateoly from the definitions of the concepts. so review those definitions.
 

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