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Thread: Approximate eigenvalues

  1. #1 Approximate eigenvalues 
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    This is a concept I had not encountered before, and I am worried by it - and slightly ashamed to have to seek help.

    Start here. (I am working from a text by Paul Halmos on Hilbert spaces).

    I am reminded that the spectrum of an arbitrary operator ( is in the first instance a finite-dimensional Hilbert space) is the the set of all complex numbers for which . This comes from the classical definition: . (Actually, it is common to pop the identity operator as factor in the first equality, but that, I believe, is merely to simplify subsequent calculations).

    First question: Halmos now writes this for the general case as by insisting that be a unit vector. Sorry but I cannot see how using a unit vector allows me move from the determinant to the norm.

    Halmos then introduces the concept of an approximate eigenvalue by for some non-negative number and still a unit vector.

    Second question: Why on earth might one be interested in "approximate" eigenvalues?

    Finally this: I am told that the last inequality is equivalent to for any arbitrary vector . Insult to injury is added by his remark that this is "easy to verify".

    I guess I need some help here too.


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  3. #2 Re: Approximate eigenvalues 
    . DrRocket's Avatar
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    Quote Originally Posted by Guitarist
    This is a concept I had not encountered before, and I am worried by it - and slightly ashamed to have to seek help.

    Start here. (I am working from a text by Paul Halmos on Hilbert spaces).

    I am reminded that the spectrum of an arbitrary operator ( is in the first instance a finite-dimensional Hilbert space) is the the set of all complex numbers for which . This comes from the classical definition: . (Actually, it is common to pop the identity operator as factor in the first equality, but that, I believe, is merely to simplify subsequent calculations).

    First question: Halmos now writes this for the general case as by insisting that be a unit vector. Sorry but I cannot see how using a unit vector allows me move from the determinant to the norm.

    Halmos then introduces the concept of an approximate eigenvalue by for some non-negative number and still a unit vector.

    Second question: Why on earth might one be interested in "approximate" eigenvalues?

    Finally this: I am told that the last inequality is equivalent to for any arbitrary vector . Insult to injury is added by his remark that this is "easy to verify".

    I guess I need some help here too.
    Determinants are defined on for operators on a finite-dimensional space. The norm applies to any normed space so in particular to any Hilbert space.

    Read on to see why approximate eigenvectors are of interest.



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