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Math Help - Matrices, Square matrix

  1. #1
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    Matrices, Square matrix

    (a) Prove that the transpose of the sum of two matrices A and B is the sum of their transposes.




    (b) Verify this is so when
    A = [1 2 3] and B = [4 6 8]
    [4 5 6] [5 7 9]




    (c) Prove that if A is a square matrix then S = 1/2 (A -A^t) is skew-symmetric.






    (d) Verify that this S is skew-symmetric when A = [4 3]
    [2 1]



    (e) Prove that if A is a square matrix with complex elements, then H = 1/2 (A + A^*) is hermitian.







    (f) Verify that this H is hermitian when A = [1+i 4-3i]


    [2-i 3+2i]






    Any help with any of these questions would be greatly appreciated!

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  2. #2
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    Transpose of a matrix

    Hello mr_motivator
    Quote Originally Posted by mr_motivator View Post
    (a) Prove that the transpose of the sum of two matrices A and B is the sum of their transposes.

    (b) Verify this is so when
    A = [1 2 3] and B = [4 6 8]
    [4 5 6] [5 7 9]

    (a) Denote the elements in row i, column j of the matrices A and B by a_{i,j} and b_{i, j}. In the matrix sum A+B, the element in row i, column j is therefore a_{i,j} + b_{i, j}.

    This is then the element in row j, column i of (A+B)^T, which is the sum of elements in row j, column i of A^T and B^T.

    This is true for all valid i and j. Hence (A+B)^T=A^T +B^T.

    (b) is a simple matter of checking this out.
    <br />
A+B = \begin{pmatrix}5 & 8 & 11\\9 & 12 & 15\end{pmatrix}

    \Rightarrow (A+B)^T =\begin{pmatrix}5 & 9\\8 & 12\\11 & 15\end{pmatrix}

    = etc...

    Grandad
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  3. #3
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    (c) Prove that if A is a square matrix then S = 1/2 (A -A^t) is skew-symmetric.
    You have to prove: S^T = -S

    Use these 3 properties:
    • (cA)^T = cA^T
    • (A+B)^T = A^T + B^T
    • \left(A^T\right)^T = A


    So:
    \begin{aligned} S^{T} & = \left[\frac{1}{2} \left(A - A^{T}\right)\right]^{T} \\ & = \frac{1}{2}\left(A - A^T\right)^{T} \\ & = \cdots \end{aligned}

    I'm sure you can finish off.

    For (d), just show that: A^T = -A
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  4. #4
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    Hermitian Matrices

    Hello mr_motivator
    Quote Originally Posted by mr_motivator View Post
    (e) Prove that if A is a square matrix with complex elements, then H = 1/2 (A + A^*) is hermitian.


    (f) Verify that this H is hermitian when A = [1+i 4-3i]


    [2-i 3+2i]
    A matrix is Hermitian if its transpose is also its conjugate. In other words, if (a_{i,j})^* = a_{j, i} for all valid i, j. We have to prove that

     \tfrac{1}{2}(A + (A^T)^*) is Hermitian.


    Now suppose in matrix A, a_{i, j} is written p + iq and a_{j,i} is written r + is. Then the element in row i, column j of \tfrac{1}{2}(A + (A^T)^*) is


     \tfrac{1}{2}(a_{i,j} + (a_{j,i})^*) = \tfrac{1}{2}[(p+iq)+(r-is)]= \tfrac{1}{2}[(p+r) +(q-s)i]
    (1)

    and the element in row j, column i of \tfrac{1}{2}(A + (A^T)^*) is


     \tfrac{1}{2}(a_{j,i} + (a_{i,j})*) = \tfrac{1}{2}[(r+is)+(p-iq)]= \tfrac{1}{2}[(p+r) +(s-q)i]

    which is the conjugate of the element in (1).

    This then is the required proof.

    I think you can complete part (f) now.

    Grandad
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