Science > Physics > Quantum Gravity 142.5: Eigenvalue Invariants As Maximum-Proximity Elements
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Science > Physics |
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"OsherD" |
| Date: |
19 May 2007 09:13:23 AM |
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Quantum Gravity 142.5: Eigenvalue Invariants As Maximum-Proximity Elements |
From Osher Doctorow
The invariant of a matrix can be defined as a function of its
underlying linear transformation which doesn't change under change of
basis. The main invariants of an n x n matrix A are det(A), tr(A),
and its eigenvalues, although such things as the rank of a matrix
(dimenson of the kernel) are technically invariant. I will take the
matrix as real (having real elements) here.
Diagonal and triangular matrices just have their eigenvalues as the
diagonal elements, and so the previous post on diagonal elements
having maximum "self-proximity" applies.
Square matrices whose eigenvectors are distinct or which span the
associated vector space of linear transformations or are independent
are diagonalizable (similar to diagonal matrices), and so with a few
modifications and conditions the maximum proximtiy idea is quite
generalizable to most real square matrices used in physics.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity 142.5: Eigenvalue Invariants As Maximum-Proximity Elements |
19 May 2007 09:20:36 AM |
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From Osher Doctorow
I meant to type rank (dimension of range), not rank (dimension of
kernel).
Osher
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