- #1

Gale

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ok, so here's the deal. The final was a take home, and an intended "learning tool." meaning, 2 weeks ago he threw us a packet of notes, said class was over, and good luck with the final. I hate this prof... somehow he's well respected in the dept... anyways, he doesn't teach us at all, this whole semesmter has basically been self taught, but whatever... the final's not too hard... just... weird... i hate his tests...

So, i'll write the questions, write what i have, and hopefully you all will help.

Also, he wants us to type these all up, and I'm not sure how to go about that... i don't know how to type matrices in word... so any advice as to how to do that is appreciated as well...

therefore [tex] X=\xi[/tex] and X is an eigenvector of A with a coresponding eigenvalue of a.

(i'm not sure if that proves "

where [tex] U^T U= I[/tex] because U is orthagonal, and therefore symmetric.

[tex]\lambda (X,Y) = (\lambda X,Y) = (AX,Y) = (X,AY) = (X,\mu Y) = \mu (X,Y) [/tex] and since [tex]\mu \neq \lambda[/tex] then we must have [tex] (X,Y)=0 [/tex]

(ok, i pretty much copied that directly from those notes he gave us... not sure if its right... or if it is, why he'd put it on the final... the guy makes no sense... )

[tex]\left(\begin{array}{cccc}0&0&0&-1\\0&1&\sqrt{2}&0\\0&\sqrt{2}&1&0\\-1&0&0&-1\end{array}\right) [/tex]

find an orthagonal matrix U so that [tex] U^{-1} AU[/tex] is a diagonal matrix.

[tex]\left(\begin{array}{ccc}1&0&-1\\0&2&0\\-1&0&1\end{array}\right) [/tex]

i) find the distinct eigenvalues [tex]\lambda[/tex] of A and their multiplicities

ii) Determine the dimensions of the eigenspaces [tex] N(A- \lambda I)[/tex]

iii) find orthonormal bases of these eigenspaces

iv) Combine these bases into one orthonormal basis B of [tex]R^3[/tex] and verify that the matrix of A relative to B is a diagonal matrix with entries the eigenvalues of A, each repeated as many times as its multiplicity

my first eigenvector is [tex] \xi_{1} =\left(\begin{array}{cc}1\\0\\1\end{array}\right) [/tex] so the dimension of that is 1?

the other two eigenvectors I'm not sure about. I get

[tex]\left(\begin{array}{cc}-1&-1\\0&0\\1&1\end{array}\right) [/tex] I'm not even sure if that's the right notation... or if the numbers are right, cause i had an undefined variable. but i wrote it this way so it'd have a dimension of 2... cause our teacher said there's was one dim1 one dim2

after that, I'm not really sure what to do, because we never went over orthonormal anything... and i don't know how to find bases well... its a mess... i'll have to sit with my notes for a while in order to understand this at all

but anyways, thanks for any help, or just checking my answers. And if you have an idea as to how i can type this all up, that'd be awesome. (maybe i can even print my work straight off pf? cause i don't know how else to type matrices...)

~gale~

So, i'll write the questions, write what i have, and hopefully you all will help.

Also, he wants us to type these all up, and I'm not sure how to go about that... i don't know how to type matrices in word... so any advice as to how to do that is appreciated as well...

**Q1)**explain why, if a is an eigenvalue of the nxn matrix A, ie, a root of f(t)= lA-tIl there is an [tex] X \neq 0[/tex] in [tex] R_{n} [/tex] with AX= aX.**A)**is a is an eigenvector of A then by definition it has a corresponding eigenvector [tex]\xi[/tex], not equal to zero, where [tex] A\xi= a\xi[/tex]therefore [tex] X=\xi[/tex] and X is an eigenvector of A with a coresponding eigenvalue of a.

(i'm not sure if that proves "

*exactly*when" do i need to show the reverse or something?)**Q2)**Show that if U is an nxn matrix, then (UX,UY) = (X,Y) exactly when the columns of U form an orthagonal set.**A)**[tex] (UX,UY)= (UX)^T (UY) = X^T U^T UY = X^T (U^T U) Y= X^T Y = (X,Y) [/tex]where [tex] U^T U= I[/tex] because U is orthagonal, and therefore symmetric.

**Q3)**Let C be an orthagonal nxn matrix (as in 2) and A an nxn matrix. Show that A is symmetric exactly when [tex] C^-1 AC[/tex] is symmetric.**A)**[tex] (C^{-1} AC)^T = (C^{-1} (AC))^T = (AC)^T (C^{-1})^T = C^T A^T (C^{-1})^T = C^{-1} A^T C[/tex] because [tex] C^T = C^{-1}[/tex] because U is orthagonal. therefore. [tex] (C^{-1} AC)^T [/tex] is symmetric when [tex] A= A^T [/tex], when A is symmetric..**Q4)**Let A be an upper triangular matrix with all diagonal entries equal. Show that if A is not a diagonal matrix, then A cannot be diagonalized.**A)**A = aI + B... (i don't know what diagonal means... so I'm not really sure what to do here... help)**Q5)**Show that A is a symmetric nxn matrix and Y and X are eigenvectors of A belonging to different eigevalues, then (X,Y)=0**A)**[tex] AX=\lambda X[/tex] and [tex] AY=\mu Y[/tex] where [tex] \lambda \neq \mu [/tex][tex]\lambda (X,Y) = (\lambda X,Y) = (AX,Y) = (X,AY) = (X,\mu Y) = \mu (X,Y) [/tex] and since [tex]\mu \neq \lambda[/tex] then we must have [tex] (X,Y)=0 [/tex]

(ok, i pretty much copied that directly from those notes he gave us... not sure if its right... or if it is, why he'd put it on the final... the guy makes no sense... )

**Q6)**Let A be the matrix[tex]\left(\begin{array}{cccc}0&0&0&-1\\0&1&\sqrt{2}&0\\0&\sqrt{2}&1&0\\-1&0&0&-1\end{array}\right) [/tex]

find an orthagonal matrix U so that [tex] U^{-1} AU[/tex] is a diagonal matrix.

**A)**ok... this one took a page and a half of work just to find one eigenvector... i have to find 4 i believe. The first eigenvector i got was [tex]\left(\begin{array}{cccc}0\\1\\1\\0\end{array}\right) [/tex] if someone could check that... that'd be cool. After i find the eigenvectors i... umm... have no idea what i do actually... combine them into one matrix? is that U? not too sure... anyways, if its necessary i'll show more of my work, I'm just not having fun will all this tex right now...**Q7)**Let A be the matrix[tex]\left(\begin{array}{ccc}1&0&-1\\0&2&0\\-1&0&1\end{array}\right) [/tex]

i) find the distinct eigenvalues [tex]\lambda[/tex] of A and their multiplicities

ii) Determine the dimensions of the eigenspaces [tex] N(A- \lambda I)[/tex]

iii) find orthonormal bases of these eigenspaces

iv) Combine these bases into one orthonormal basis B of [tex]R^3[/tex] and verify that the matrix of A relative to B is a diagonal matrix with entries the eigenvalues of A, each repeated as many times as its multiplicity

**A)**ok [tex] \lambda_{1} = 0 \lambda_{2,3} = 2[/tex]my first eigenvector is [tex] \xi_{1} =\left(\begin{array}{cc}1\\0\\1\end{array}\right) [/tex] so the dimension of that is 1?

the other two eigenvectors I'm not sure about. I get

[tex]\left(\begin{array}{cc}-1&-1\\0&0\\1&1\end{array}\right) [/tex] I'm not even sure if that's the right notation... or if the numbers are right, cause i had an undefined variable. but i wrote it this way so it'd have a dimension of 2... cause our teacher said there's was one dim1 one dim2

after that, I'm not really sure what to do, because we never went over orthonormal anything... and i don't know how to find bases well... its a mess... i'll have to sit with my notes for a while in order to understand this at all

but anyways, thanks for any help, or just checking my answers. And if you have an idea as to how i can type this all up, that'd be awesome. (maybe i can even print my work straight off pf? cause i don't know how else to type matrices...)

~gale~

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