Symmetric Matrices and Manifolds Answer Guide

In summary, the conversation involves proving that (A^T)A is symmetric for an n x n matrix A, showing that the set of symmetric n x n matrices is a subspace of the set of all n x n matrices, determining the dimension of this set, and computing the derivative of the function f(X)=(X^T)X-I. It is also shown that the tangent space to the set of orthogonal matrices at the identity matrix is the space of skew-symmetric matrices, and this is the same dimension as the manifold of orthogonal matrices.
  • #1
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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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  • #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?
 
  • #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?
 
  • #4
the first few of these problems follow immediateoly from the definitions of the concepts. so review those definitions.
 

FAQ: Symmetric Matrices and Manifolds Answer Guide

1. What is a symmetric matrix?

A symmetric matrix is a square matrix where the values are mirrored across the main diagonal, creating a reflection symmetry. This means that the value at position (i,j) is equal to the value at position (j,i).

2. What are some properties of symmetric matrices?

Some properties of symmetric matrices include:

  • They are always square matrices.
  • They are equal to their own transpose.
  • They have real eigenvalues.
  • Their eigenvectors are orthogonal.

3. How are symmetric matrices used in mathematics and science?

Symmetric matrices are used in various fields of mathematics and science, including:

  • Linear algebra: They are used to represent quadratic forms and are important in the spectral theorem.
  • Statistics: They are used in multivariate analysis and correlation matrices.
  • Physics: They are used in quantum mechanics to represent physical observables.
  • Computer science: They are used in machine learning algorithms, such as principal component analysis.

4. What is a manifold?

A manifold is a mathematical concept used to describe a space that locally resembles Euclidean space. It is a topological space that is locally homeomorphic to Euclidean space, meaning that it can be stretched and bent, but not torn or glued together.

5. How are symmetric matrices related to manifolds?

Symmetric matrices are closely related to manifolds, as they can be used to define a Riemannian metric on a manifold. This metric allows for the measurement of distances and angles on the manifold, and is used in many fields including differential geometry, relativity, and optimization. Symmetric matrices also play a role in the study of geodesics on manifolds, which are the shortest paths between two points.

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