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Topic: Failing Linear Algebra:
Replies: 54   Last Post: Jan 10, 2007 12:47 PM

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Law Hiu Chung

Posts: 303
Registered: 12/7/04
Re: Failing Linear Algebra:
Posted: Apr 28, 2004 7:03 PM
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Anonymous wrote:
> Grey Knight:

>>If you calculate det(<>-l*<>) for an n-by-n matrix <>, you
>>will get a degree-n polynomial in l; generally you'll be expected to
>>deal with 2-by-2 matrices, so this polynomial is just a quadratic. The
>>solutions of this polynomial are your eigenvalues.


> Thanks, Grey Knight. I've got that part down.

>>The eigenvectors
>>can then be found by solving the problem <> <x_i> = l_i <x_i>; that
>>is (<> - l_i<>)<x_i> = <0>


> This is the part I don't get. I'm sorry but your notation loses me. I see the
> "<>", and I think "dot product".


> We find the eigenvalues by solving matrix A....det (A-[lambda]*I ), where I is
> the identity matrix.


What Gary is saying is this.... Suppose that A is a 2 by 2 matrix

(a b)
(c d)

and then lambda is the eigenvalue. You should then solve for x1 and x2 in

(a-lambda b ) (x1) = (0)
(c d-lambda) (x2) (0)

Although there are two equations here, one of them is redundant. So you
can just solve either one... Since you have one equation, two unknowns,
there can be infinite solutions on x1 and x2. For simplicity, just pick
x1=1 and try to solve x2. If you can solve it, then you are done,
after you normalize the eigenvectors to norm one. If you get contradiction
by assuming x1=1, then x1 should be 0.

>>Two matrices are similar iff
>> <> = <<X>> <> <<X'>>


> OK. So I = X*I*X'. So, X' is any matrix that when multiplied by X gives the
> identity element as well?


Actually... I think there is a mistake here.

A n by n matrix X is similar to a n by n Y if you can find a n by n matrix T
such that

X = T * Y * inv(T)

where inv(T) denotes the inverse of T, inv(T) * T = I

Inverse and transpose are two different concepts

If A is

1 2
3 4

then A' (A transpose) is

1 3
2 4

and A inverse is

( 4 -3 )
( -2 1 ) / (-2)


> What's the difference between A transpose and A inverse? A specific example(s)
> would help. I'd love to test my skills.


Good luck to your exam.


Date Subject Author
4/22/04
Read Failing Linear Algebra:
Guest
4/22/04
Read Re: Failing Linear Algebra:
Michael N. Christoff
1/10/07
Read Re: Failing Linear Algebra:
Gerry Myerson
1/10/07
Read Re: Failing Linear Algebra:
Jonathan Miller
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich
1/10/07
Read Re: Failing Linear Algebra:
Acid Pooh
1/10/07
Read Re: Failing Linear Algebra:
Guest
4/23/04
Read Re: Failing Linear Algebra:
Russell Blackadar
4/23/04
Read Re: Failing Linear Algebra:
Brian Borchers
4/27/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
maky m.
4/26/04
Read Re: Failing Linear Algebra:
David Ames
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Michael Stemper
1/10/07
Read Re: Failing Linear Algebra:
maky m.
4/23/04
Read Re: Failing Linear Algebra:
Porker899
4/27/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Abraham Buckingham
1/10/07
Read Re: Failing Linear Algebra:
Mitch Harris
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Grey Knight
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Toni Lassila
1/10/07
Read Re: Failing Linear Algebra:
Thomas Nordhaus
1/10/07
Read Re: Failing Linear Algebra:
George Cox
4/28/04
Read Re: Failing Linear Algebra:
Dave Rusin
4/28/04
Read Re: Failing Linear Algebra:
George Cox
4/28/04
Read Re: Failing Linear Algebra:
George Cox
4/29/04
Read Re: Failing Linear Algebra:
Marc Olschok
4/29/04
Read Re: Failing Linear Algebra:
Mitch Harris
4/29/04
Read Re: Failing Linear Algebra:
Robert Israel
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/28/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/29/04
Read Re: Failing Linear Algebra:
Russell Blackadar
4/29/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
5/1/04
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
1/10/07
Read Re: Failing Linear Algebra:
Dave Rusin
1/10/07
Read Re: Failing Linear Algebra:
Russell Blackadar
4/30/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich
4/27/04
Read Re: Failing Linear Algebra:
Guest
4/27/04
Read Re: Failing Linear Algebra:
Guest
4/28/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
Law Hiu Chung
4/30/04
Read Re: Failing Linear Algebra:
Guest
1/10/07
Read Re: Failing Linear Algebra:
David C. Ullrich

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