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Math Help - Linear Algebra tough ones

  1. #1
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    Linear Algebra tough ones

    Consider the vector space M_{2} (C) of all 2x2 matrices with complex entries. If A= \left(\begin{array}{cc}a&b\\c&d\end{array}\right) then A* denotes the conjugate transpose of A.

    A.) For A,B belong to M_{2} (C), show that <A,B>=tr(AB*) defines an inner product.

    B.) Find an orthonormal basis for M_{2} (C) with respect to this inner product.
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  2. #2
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    Quote Originally Posted by mathisthebestpuzzle View Post
    Consider the vector space M_{2} (C) of all 2x2 matrices with complex entries. If A= \left(\begin{array}{cc}a&b\\c&d\end{array}\right) then A* denotes the conjugate transpose of A.

    A.) For A,B belong to M_{2} (C), show that <A,B>=tr(AB*) defines an inner product.

    B.) Find an orthonormal basis for M_{2} (C) with respect to this inner product.
    A) Could you show us your work? After all, you have to just verify some rules. Where are you stuck?

    B) Did you try Gram-Schmidt?
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  3. #3
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    For A.) i had to check linearity in the first slot, positive definiteness, as well as conjugate symmetry of the inner product.

    i got stuck here, showing the positive definiteness.

    B.) Is \left(\begin{array}{cc}1&0\\0&0\end{array}\right), \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right),
    \left(\begin{array}{cc}0&0\\0&1\end{array}\right) an orthonormal basis for M_2(C) ? and if not why?!
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  4. #4
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    Hi
    Quote Originally Posted by mathisthebestpuzzle View Post
    For A.) i had to check linearity in the first slot, positive definiteness, as well as conjugate symmetry of the inner product.

    i got stuck here, showing the positive definiteness.
    \mathrm{tr}(A \overline{^tA})=\mathrm{tr} \left( \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} \overline{a} & \overline{c}\\ \overline{b} & \overline{d} \end{pmatrix} \right)=\mathrm{tr} \left( \begin{pmatrix} a\overline{a}+b\overline{b} & a\overline{c}+b\overline{d}\\ c\overline{a}+d\overline{b} & c\overline{c}+ d\overline{d} \end{pmatrix} \right)
    \mathrm{tr}(A \overline{^tA})=a\overline{a}+b\overline{b}+c\over  line{c}+ d\overline{d}

    What can be said about that ?
    B.) Is \left(\begin{array}{cc}1&0\\0&0\end{array}\right), \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right),
    \left(\begin{array}{cc}0&0\\0&1\end{array}\right) an orthonormal basis for M_2(C) ? and if not why?!
    How do you get \begin{pmatrix} \imath & 0\\ 0 & \imath\end{pmatrix} using this basis ? <- wrong

    To apply Gram-Schmidt, you need a basis of  \mathcal{M}_2(\mathbb{C}) which is not necessarily orthonormal, to start the "process".
    Last edited by flyingsquirrel; April 26th 2008 at 02:56 PM.
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  5. #5
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    for A, i got to the place where you did, with the trace. I am unsure of how to show that this will always be positive.

    For B, could i start with a the basis i described but instead of 1's use i? and then apply the gram schmidt?
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by mathisthebestpuzzle View Post
    for A, i got to the place where you did, with the trace. I am unsure of how to show that this will always be positive.
    ... z\in\mathbb{C} \Rightarrow z\overline{z}=?
    For B, could i start with a the basis i described but instead of 1's use i? and then apply the gram schmidt?
    I was wrong, (and need to go to bed ) the first basis you chose was correct : you can apply Gram-Schmidt to it...
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  7. #7
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    much appreciated.
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  8. #8
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    how does one apply the gramm schmidt to a basis of matrices?
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  9. #9
    Super Member flyingsquirrel's Avatar
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    In fact, it seems that the basis you chose is already orthonormal. You should check by computing \mathrm{tr}(E_i E_j) : it should equal 1 if i=j ("normal") and 0 otherwise. ("ortho")

    with (E_1, E_2, E_3, E_4)= \left(\left(\begin{array}{cc}1&0\\0&0\end{array}\r  ight), \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right),<br />
\left(\begin{array}{cc}0&0\\0&1\end{array}\right)\  right)
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  10. #10
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    funny that i checked this right now, i'm half way through the gram-schmidt process and the process isn't doing anything....lol. thank you flying squirrel.
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  11. #11
    Super Member flyingsquirrel's Avatar
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    It does not do anything because there is nothing to do as the basis is already orthonormal.
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  12. #12
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    part C. U is the subspace of M_2(C) of matrices A with tr(A)=0. Find an orthonormal basis of U and describe U perp.

    I think that a basis of U could be:

      \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right),<br />
\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)

    and i can make it an orthonormal basis using the gram schmidt right?

    how would i describe U perp?
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  13. #13
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by mathisthebestpuzzle View Post
    part C. U is the subspace of M_2(C) of matrices A with tr(A)=0. Find an orthonormal basis of U and describe U perp.

    I think that a basis of U could be:

      \left(\begin{array}{cc}0&1\\0&0\end{array}\right), \left(\begin{array}{cc}0&0\\1&0\end{array}\right),<br />
\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)

    and i can make it an orthonormal basis using the gram schmidt right?
    I think your basis is already orthonormal.
    how would i describe U perp?
    The dimension of U is three so either U^{\perp}=\{0\}, either the dimension of U^{\perp} is 1. To find a basis of U^{\perp}, you can take a random matrix which trace does not equal 0 (say I_2 for example) and see how you can express it has a sum of matrices of U and of one matrix which is not in U. Then, check if this last matrix is in U^{\perp} and... I let you take it from here.
    Last edited by flyingsquirrel; April 29th 2008 at 10:02 AM. Reason: matrix -> matrices :/
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