# HG changeset patch # User Rob Beezer # Date 1354731161 28800 # Node ID d6033fe320b0e6af092c028c7791d20db1f542d7 # Parent b339151ce840e858ab5fc016dd08f21ecc6215c2 Rewrote discussion between Theorem EIM and Theorem ETM (Kevin Halasz) diff --git a/changes.txt b/changes.txt --- a/changes.txt +++ b/changes.txt @@ -22,6 +22,7 @@ Edit: Solution SLT.C29, added solution to surjective question (Dan Messenger) Edit: Removed two forward references to Chapter MD (Aidam Meacham) Edit: Proof of Theorem NME7 clarified (Kevin Halasz) +Edit: Rewrote discussion between Theorem EIM and Theorem ETM (Kevin Halasz) Change: Replaced Definition RR by an informal discussion Change: Definition TSVS renamed as Definition SSVS Change: Proof Technique D: discussion about definitions as nouns, adjectives diff --git a/src/section-PEE.xml b/src/section-PEE.xml --- a/src/section-PEE.xml +++ b/src/section-PEE.xml @@ -365,7 +365,10 @@ -

The theorems above have a similar style to them, a style you should consider using when confronted with a need to prove a theorem about eigenvalues and eigenvectors. So far we have been able to reserve the characteristic polynomial for strictly computational purposes. However, the next theorem, whose statement resembles the preceding theorems, has an easier proof if we employ the characteristic polynomial and results about determinants.

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The proofs of the theorems above have a similar style to them. They all begin by grabbing an eigenvalue-eigenvector pair and adjusting it in some way to reach the desired conclusion. You should add this to your toolkit as a general approach to proving theorems about eigenvalues.

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So far we have been able to reserve the characteristic polynomial for strictly computational purposes. However, sometimes a theorem about eigenvalues can be proved easily by employing the characteristic polynomial (rather than using an eigenvalue-eigenvector pair). The next theorem is an example of this.

+ Eigenvalues of the Transpose of a Matrix