posted by .

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

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???

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. spanish

    What is the rule for formal commands? This site give information on Direct Commands. This site shows the difference between commands and requests. It also gives the verb forms:
  2. math

    Prove that if A is a diagonalizable matrix, then the rank of A is the number of nonzero eigenvalues of A. I've read the entire page and while it's on the …
  3. please help me(math)

    4x^2+20x+5y+xy *you have top factor by grouping.If you need more info just ask me but please help me!!!* An example from Factoring by Grouping Consider the expression …
  4. science

    how many groups of birds are there and name them Since this is not my area of expertise, I searched Google under the key words " bird classification " to get these possible sources: …
  5. IGCSE Biology

    Okay i have an exam in a couple of weeks, and i don't know how birds reproduce. So i was wondering whether or not anyone knew the detailed steps for it/ how it happens. thanx for your help!!! :D Birds reproduce by sexual reproduction, …
  6. (Life)

    What is a good conflict resolution technique for employees?
  7. Math

    How do you remember all the regruping stuff love Ally pethebridge First, for your own safety, use only your first name in posting. Thanks. Remembering "regruping stuff" is accomplished by the same methods used for studying any subject. …
  8. Science

    Please help, I am so bored by my science lesson that I can barely focus. I read the lesson, and 5 seconds later I don't even remember what I just read. Any ideas on somehow making the Atmosphere and Solar Effects more interesting?
  9. Health

    What are the requirements for the career of medical assistant?
  10. Biology

    Explain how the digestive, cardiovascular, and respiratory systems work together to allow cellular respiration to occur. Here are some good sources on cellular respiration: …

More Similar Questions