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Math Help - Unitary and triangular = diagonal matrix

  1. #1
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    Unitary and triangular = diagonal matrix

    can anybody help me with this question?

    Show that if a matrix A is both triangular and unitary, then it is diagonal.

    thanks
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  2. #2
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    Quote Originally Posted by Bucephalus View Post

    can anybody help me with this question?

    Show that if a matrix A is both triangular and unitary, then it is diagonal.

    thanks
    i'll assume that the matrix A=[a_{ij}], \ 1 \leq i,j \leq n, is upper triangular. the lower triangular case is the same. proof is by induction over n: for n=2 it's easy. now if [b_{ij}]=AA^*=I_n, where A^*

    is the complex conjugate of A, then we'll have b_{in}=0, \ \forall \ i \leq n-1, but b_{in}=a_{in} \overline{a_{nn}} and a_{nn} \neq 0. therefore b_{in}=0, \ \forall \ i \leq n-1. now apply the induction hypothesis for the (n-1) \times (n-1)

    matrix C=[a_{ij}], \ 1 \leq i,j \leq n-1 to finish the proof.
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  3. #3
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    then we'll have b_{in}=0, \ \forall \ i \leq n-1, but b_{in}=a_{in} \overline{a_{nn}} and a_{nn} \neq 0. therefore b_{in}=0, \ \forall \ i \leq n-1.


    I don't really understand how

    b_{in}=a_{in} \overline{a_{nn}} and a_{nn} \neq 0.
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  4. #4
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    b_{in}=a_{in} \overline{a_{nn}} and a_{nn} \neq 0.
    From what I'm reading here is that for a say a 2X2 matrix, the element
    b_{i2} is the a_{i2} element multiplied by the
    b_{22} element.

    Is that correct?
    i'm going to try that on paper.
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  5. #5
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    SOrry, I mean the conjugate of the a_{22} element.
    not b_{aa} element.

    Don't worry,I will do some paper work here.
    I think I'm starting to understand some of your proof using a 2X2 Identity matrix.

    thanks for your response.
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