math

Prove that if A is a symmetric n x n matrix, then A has a set of n orthonormal eigenvectors.

http://ltcconline.net/greenl/courses/203/MatrixOnVectors/symmetricMatrices.htm

I've read the entire page and while it's on the correct topic, it doesn't prove what I'm looking to prove.

I think you want proof of the "completeness" property, i.e. that there are n orthonormal eigenvectors, not that if there are two eigenvectors then they must be orthogonal (or in the degenerate case that you can choose orthonormal eigenspaces corresponding to different eigenvalue).

Put differently this means that the eigenvectors span the entire linear space the matrix is acting on.

You can prove this using induction. If an n-by n matrix A has one eigenvector V with eigenvalue lambda, then you consider the linear operator defined as:

A dot x - lambda (V dot x) V

Here x is a vector on which we let the operator act on. A dot x is the action of A on x. The second term is the inner product of x with V we then multiply that by the vector V times lambda. This is just like what the matrix A does with V. By subtracting this term you map V to zero.

This means that the linear operator you obtained maps the orthogonal complement of vectors proportional to V, which is an n-1 dimensional space onto itself.

You can iterate this procedure until you encounter the trivial case of a linear operator acting on a 1 dimensional space.

All you need in this proof is the fact that a symmetric linear operator has at least one eigenvector.

can an inner product space v have a t invariant subspace U but also have an orthogonal complement that is NOT t-invariant???

  1. 👍 0
  2. 👎 0
  3. 👁 183

Respond to this Question

First Name

Your Response

Similar Questions

  1. science

    1 A ……... is a rectangular array of numbers that are enclosed within a bracket . horizontal set vertical matrix 2 When the numbers of rows is equal to the numbers of columns equal to 'n'. Where m=n. Then is

    asked by sunday on March 15, 2014
  2. Math

    Mark each of the following True or False. ___ a. All vectors in an orthogonal basis have length 1. ___ b. A square matrix is orthogonal if its column vectors are orthogonal. ___ c. If A^T is orthogonal, then A is orthogonal. ___

    asked by Melissa on January 2, 2011
  3. linear algebra

    3. Suppose A is symmetric positive definite and Q is an orthogonal matrix (square with orthonormal columns). True or false (with a reason or counterexample)? a) (Q^(T))AQ is a diagonal matrix b) (Q^(T))AQ is a symmetric positive

    asked by michael on December 1, 2010
  4. math

    A matrix A is said to be skew symmetric if A^T = -A. Show that is a matrix is skew symmetric then its diagonal entries must all be 0. A^T meant to be A transpose.

    asked by kevin on January 11, 2011
  5. MATHS----Matrix

    For a given square matrix A the predicted values of matrix B are: predicted B=A(A'A)^(-1)A'B why is the matrix C=A(A'A)^(-1)A' an idempotent and symmetric matrix? and is this matrix invertible?

    asked by Lisa on September 8, 2010
  6. linear algebra

    If A is an n × n matrix, then A = S + K, where S is symmetric and K is skew symmetric. Let A= [1 3 -2;4 2 2;5 1 2] Find the matrices S and K described above can some0ne explain how to get these two matricies? thanks

    asked by sam on January 24, 2015
  7. algebra

    Hi guys: Can any one please tell me what does this means? Thanks ---------------------------------------------------------------------- The second matrix is simply the symmetric version of the first. This 1 2 3 4 5 6 7 1 0 2 3 x 2

    asked by Kathleen on December 13, 2011
  8. linear algebra

    1)If A is an invertible matrix and k is a positive integer, then (A^k)^-1 = (A^-1)^k note: ^ stand for power, -1 stand for inverse of A 2)If A is an invertible matrix, then (transpose of A)^-1 =transpose of(A^-1) 3)Prove A^2 = A,

    asked by soasi piutau on April 13, 2012
  9. Math

    If A is a square matrix, show that B=(A+A^T)/2 is a symmetric matrix.

    asked by Jay on September 25, 2016
  10. math

    Hi, I just want to make sure I am doing this right: Construct a relation on the set {a, b, c, d} that is a) reflexive, symmetric, but not transitive. b) irrreflexive, symmetric, and transitive. c) irreflexive, antisymmetric, and

    asked by Confused!! on March 15, 2012

More Similar Questions