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Math Help - Frobenius inner product

  1. #1
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    Frobenius inner product

    Use the Frobenius inner product to compute ||A||, ||B||, and <A,B> for

    A= (1, 2+i,3,i) (A is supposed to be a matrix so a_11=1, a_12=2+i, a_21=3,a_22=i)

    and B=(1+i, 0, i,-i) (again B is supposed to be a matrix so b_11=1+i, b_12=0, b_21=i, b_22=-i)

    I'm not familiar with Frobenius inner product, so im not sure how to do this problem.
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  2. #2
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    Quote Originally Posted by alice8675309 View Post
    Use the Frobenius inner product to compute ||A||, ||B||, and <A,B> for

    A= (1, 2+i,3,i) (A is supposed to be a matrix so a_11=1, a_12=2+i, a_21=3,a_22=i)

    and B=(1+i, 0, i,-i) (again B is supposed to be a matrix so b_11=1+i, b_12=0, b_21=i, b_22=-i)

    I'm not familiar with Frobenius inner product, so im not sure how to do this problem.

    Go to the LaTeX help section in the main page to learn how to properly write mathematics, in particular to write matrices, so that you can make your point crystal clear.

    So A=\begin{pmatrix}1&2+i\\3&i\end{pmatrix}\,,\,\,B=\  begin{pmatrix}1+i&0\\i&-i\end{pmatrix} \Longrightarrow A^*=\overline{A^t}=\begin{pmatrix}1&3\\2-i&-i\end{pmatrix}\,,\,\,B^*=\overline{B^t}=\begin{pma  trix}1-i&-i\\0&-i\end{pmatrix} , so:

    \langle A,B\rangle:=tr.(AB^*)=tr.\begin{pmatrix}1-i&1-3i\\3-3i&1-3i\end{pmatrix} , and also ||A||:=\langle A,A\rangle , so now you can do the maths.

    Tonio
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  3. #3
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    Isn't it normally

    \|A\|:=\sqrt{\langle A,A\rangle}?
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  4. #4
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    Quote Originally Posted by Ackbeet View Post
    Isn't it normally

    \|A\|:=\sqrt{\langle A,A\rangle}?

    Of course youŽre right. Thanx

    Tonio
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    Isn't it normally

    \|A\|:=\sqrt{\langle A,A\rangle}?
    Quick question: With the last part of alice8675309's question, how do you take the inner product of two matrices. Namely <A,A>. I completely forgot.
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  6. #6
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    I started by opening my Linear Algebra book and then said:

    \displaystyle <A,B>=\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}b_{ij}

    and

    \displaystyle <A,A>=\sum_{i=1}^{m}\sum_{j=1}^{n}(a_{ij})^2
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