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Math Help - Matrix Linear Equations

  1. #1
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    Matrix Linear Equations

    Anyone able to help me verify?

    Consider a sequence of real-number r1, r2, ....,rN
    Using this we create an (N * N) square matrix R such that the (i, j)th element of R is given by rK , where k = min(i,j), i = 1, 2, ...., N ; j = 1, 2, .....,N

    A. Write the matrix R in terms of its elements. Clearly, show at least the top 3 3 part
    and all the elements on the four corners.

    r11 r1 2 r13 ------ r1N * x1 = b1
    r21 r22 r23----- -r2N * x2 = b2
    r31 r32 r33--------r3N * x3 = b3
    ri1 ri2 ri3-------riN * xj = bj
    rm1 rm2 rm3------ rmn * xN = bN

    B. Is this a symmetric matrix?
    yes?
    A = A^T

    Regards & a million thanks
    Chris
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  2. #2
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    Quote Originally Posted by Chris0724 View Post

    Consider a sequence of real-number r1, r2, ....,rN
    Using this we create an (N * N) square matrix R such that the (i, j)th element of R is given by rK , where k = min(i,j), i = 1, 2, ...., N ; j = 1, 2, .....,N

    A. Write the matrix R in terms of its elements. Clearly, show at least the top 3 3 part
    and all the elements on the four corners.


    B. Is this a symmetric matrix?
    the matrix R is obviously symmetric because \min(i,j)=\min(j,i). here's the matrix: R=\begin{pmatrix} r_1 & r_1 & r_1 & . & . & . & r_1 \\ r_1 & r_2 & r_2 & . & . & . & r_2 \\ r_1 & r_2 & r_3 & . & . & . & r_3 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ r_1 & r_2 & r_3 & . & . & . & r_N \end{pmatrix} . can you show that \det R=r_1 \prod_{j=1}^{N-1} (r_{j+1}-r_j) ?
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  3. #3
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    thanks for the help

    i can't seem to understand the meaning where k = min(i,j), i = 1, 2, ....N; j = 1, 2,....N.

    Able to explain? thanks a million
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  4. #4
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    Hello,

    "matrix R such that the (i, j)th element of R is given by r_k , where k=\min(i, j), i = 1, 2, ...., N ; j = 1, 2, .....,N" simply means that "the (i, j)th element of R is r_{\min(i, j)} (i=1,...,N;j=1,...,N)."
    For example, the (2, 3)th element of R(usually denoted R_{23}) is r_{\min(2, 3)}. Since 2 is less than 3, \min(2, 3)=2. Thus, the (2, 3)th element of R is r_2.

    Bye
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