(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am having problems with (i)

Also I find my teacher to be very methodical, and lacking in theory. So if you find my reasoning poor or incomplete in any regard PLEASE enlighten me by clarifying. Thanks :D

12. (15 marks)

Let A =

(−1 −8 −2)

( 1 5 1)

( 0 0 1)

(a) Verify that the eigenvalues of A are 1 and 3.

(b) Find E1, the eigenspace for eigenvalue 1.

(c) Find dimE1 and a basis of E1.

(d) Find E3, the eigenspace for eigenvalue 3.

(e) Find dimE3 and a basis of E3.

(f) Determine whether A is diagonalizable. State a brief reason.

(g) If A is diagonalizable, find an invertible matrix P and a diagonal matrix D such that P^(−1)AP = D.

(h) Verify that P^(−1)AP = D is indeed true.

(i) Let B =

(2 1 1)

(1 2 1)

(0 0 1)

Is A~B? If A~B, find an invertible matrix Q such

that B = Q^(−1)AQ.

2. Relevant equations

Not sure, online based course and I find no relavent notes. Here is link: http://fsj.nlc.bc.ca/nlc/hcui/ [Broken]

Math 152, Course Material, Unit 5

3. The attempt at a solution

(a)To verify the eigenvalues, I use (tI-A) to find the characteristic equations. Since |tI-A|, I solve for t and 1 and 3 are the correct Eigenvalues.

(b-e)Then I use Av=1v and Av=3v,

where v=

(x)

(y)

(z)

Thus, (1I-A)v=0 and (3I-A)v=0,

where 0=

(0)

(0)

(0)

By forming a homogeneous matrice and putting it into RREF for (1I-A) I get

(1 4 1|0)

(0 0 0|0) Let y=r, and z=s, where r and s are parameters

(0 0 0|0)

thus E1 =

{(-4r-s) }

{( r ) :r,sE|R }

{( s ) }

and dim E1= 2, basis:

(-4)

( 1)

( 0)

and

(-1)

(0)

(1)

By forming a homogeneous matrice and putting it into RREF for (3I-A) I get

(1 2 0|0)

(0 0 1|0) Let y=t, where r is a parameter

(0 0 0|0)

thus E1 =

{( -2t ) }

{( t ) :r,sE|R }

{( 0 ) }

and dim E1= 1, basis:

(-2)

( 1)

( 0)

SO FAR, so good ;)

(f)A is diagnalizable, as the total number of basis Eigenvecrots is 3, the n dimension R space it is in. (R^3)

(g) P^(−1)AP = D

P=

(-4 -1 -2)

( 1 0 1) from Basis of eigenvector spaces

( 0 1 0)

and

D=

(1 0 0 )

(0 1 0 ) from eigenvalues corresponding to eigenvectors

(0 0 3 )

(h) I find the inverse of P, P^(-1), simply by forming an augmented matrice (p|I) then put p into RREF using row operations yielding (I|P^(-1))

P^(-1) =

(-1/2 -1 -1/2 )

( 0 0 1 )

( 1/2 2 1/2)

and upon verification P^(-1)AP=D ...sweet ^^

Here comes the problem:

(i) First of all, I'm pretty sure my rational for why A~B is complete UNTIL I can find B = Q^(−1)AQ... WHICH I'm finding questionable considering the path of questioning :(.

I know that if A~B they have equivalent determinants(check),characteristic equations(redundant check), and traces(check). HOWEVER that doesn't imply they are similar, does it? Only that it is not simple to dismiss they are similar. F my life ;[.

ANYWAYS assuming that was sufficient reasoning

I tried multiplying both sides by matrice Q yeilding

QB = AQ

(just to make it easier to follow)

Let A =

(−1 −8 −2)

( 1 5 1)

( 0 0 1)

and B =

(2 1 1)

(1 2 1)

(0 0 1)

then labeled Q the arbitrary matrice

(a b c)

(d e f)

(g h i )

multiplied both sides out and compared each entry of each matrice against each other...

*Head explodes*

After I put myself back together again(and finding out h = -g and few other very vague equations), I searched the net for help. I found a lot of advice telling me to find an invertible Q, but I am, frankly, mystified at how to approach this.

ANYTHING helpful, that you feel like posting on directly solving my problem, or areas that appear vague in my understanding will be appreciated. Please and thanks.

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# Homework Help: Linear Algebra - Similar Matrices(finding invertible Q for A~B given A and B)

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