Another Linear Algebra Proof

[ChineseGuy]

Here is another one for you smart people.

For an invertible matrix A, A and A-1 (<-- that's A inverse) have the same eigenvectors.

I am not sure where to start on this.

Thanks in advance.

 

[esun]

An eigenvector of a matrix A is a vector v such that

Av = cv

Where c is a constant (c is usually lambda, but for the sake of formatting on the forums I'm going to use c).

If A is invertible, that means we can write (I'm going to use A' to be A^-1, again for ease of formatting)

A'Av = A'cv (multiply by A' on the left side)
Iv = cA'v (scalars and vectors commute; A'A = I; also note that Iv = v)
A'v = 1/c v
A'v = kv, where k = 1/c

Therefore A and A' have the same eigenvectors, but the eigenvalues are inverted.

 

[ChineseGuy]

Thanks esun for all your help on that proof. You're a life saver.

 

[Biftheunderstudy]

Oh, but that brings back memories of Quantum Mechanics, Simultaneously Diagonalizable matrices, commuting observables.......

 

[nyker96]

damn did that in HS, tough stuff.

 

[ChineseGuy]

Another question for you guys... What book did you guys used? The text I am currently using is printed in like 1989... And its soo hard to follow. I am thinking of getting another linear algebra book to use as reference.

 

[thesurge]

Introduction to Linear Algebra-- Gilbert Strang

 

[eLiu]

Originally posted by: thesurge
Introduction to Linear Algebra-- Gilbert Strang

This text is *excellent*.

 

[ChineseGuy]

Just bought the book... It better be good! *shakes fist*

 

[imported_Seer]

While we're on the subject...did any of yall ever use Paul Foerster's textbooks in HS? For Agebra 1 and 2, pre cal, and cal? Just wondering.

 

[ChineseGuy]

This is the book we used for Calc
Linky

I don't remember the one for Algebra or Pre Calc

 

[randumb]

Linear Algebra Done Right - Sheldon Axler

This book is excellent; the exposition is clear and the proofs are clean. However, it does omit some topics like dual spaces, quotient spaces, and tensor products.

 

[mrkun]

I have a friend who thinks that Tom Apostol is the best math textbook writer in the history of human civilization. He has a Linear Algebra text.

Edit: I used Elementary Linear Algebra by Howard Anton and it was tolerable, although I'm sure better texts exist.