Results 1 to 2 of 2

Math Help - Matrices question

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    171

    Matrices question

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Matrices

    Hello champrock
    Quote Originally Posted by champrock View Post
    The answer is (c): every row and every column of P contains just one 1, and (n-1) zeros. Let me try and explain why.

    Suppose the elements of P are p_{i,j}. In other words the element in row i, column j is p_{i,j}. Suppose also that the elements of x are x_1, x_2, \dots, x_n, and the elements of y are y_1, y_2, \dots, y_n

    Then \begin{pmatrix}y_1\\y_2\\ \dots\\y_n\end{pmatrix} = \begin{pmatrix}p_{1,1}&p_{1,2}&\dots& p_{1,n}\\p_{2,1}&\dots&\dots &\dots  \\ \dots&\dots &\dots&\dots   \\p_{n,1}&\dots&\dots& p_{n,n}\end{pmatrix}\begin{pmatrix}x_1\\x_2\\ \dots\\x_n\end{pmatrix}

    Now for each i, (1\le i\le n), y_i is formed when the elements of row i of matrix P are combined (by multiplication and addition) with the elements of x. In other words:

    y_i = \sum_{k=1}^np_{i, k}x_k

    And y's elements are a re-arrangement of x's elements. So for each i, y_i = x_j for some j, 1 \le j\le n. Therefore, from the above equation:

    p_{i,k} = \left\{<br />
\begin{array}{l l}<br />
0 & \quad k \ne j\\<br />
1 & \quad k = j\\<br />
\end{array} \right.

    So each row of P contains a single 1, and (n-1) zeros. Moreover, each y_i is equal to a different x_j, so no column of P contains more than a single 1. So P is bistochastic, with elements 0 and 1.

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question about matrices
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 29th 2011, 07:54 AM
  2. a question about matrices
    Posted in the Discrete Math Forum
    Replies: 10
    Last Post: July 30th 2009, 07:32 AM
  3. matrices question
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 6th 2009, 05:33 PM
  4. Question about 3 X 3 matrices
    Posted in the Algebra Forum
    Replies: 3
    Last Post: July 24th 2008, 12:00 PM
  5. Matrices question
    Posted in the Algebra Forum
    Replies: 2
    Last Post: December 13th 2006, 02:53 AM

Search Tags


/mathhelpforum @mathhelpforum