Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The rank of a block matrix as a function of the rank of its submatrice

  1. Feb 9, 2013 #1
    Hello everyone,
    I would like to post this problem here in this forum.
    Having the following block matrix:

    [tex]
    \begin{equation}
    M=\begin{bmatrix}
    S_1 &C\\
    C^T &S_2\\
    \end{bmatrix}
    \end{equation}
    [/tex]

    I would like to find the function $f$ that holds [tex]rank(M)=f( rank(S1), rank(S2))[/tex].
    [tex]S_1[/tex] and [tex]S_2[/tex] are covariance matrices-> symmetric and positive semi-definite.
    [tex]C[/tex] is the cross covariance that may be positive semi-definite.

    Can you help me?

    I sincerely thank you! :)

    All the best

    GoodSpirit
     
    Last edited: Feb 9, 2013
  2. jcsd
  3. Feb 9, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Re: The rank of a block matrix as a function of the rank of its submat

    Are you sure that this function exists?

    [tex]
    \begin{equation}
    M=\begin{bmatrix}
    1 &1\\
    1 &1\\
    \end{bmatrix}
    \end{equation}
    [/tex]
    => rank(M)=1
    [tex]
    \begin{equation}
    M=\begin{bmatrix}
    1 &.5\\
    .5 &1\\
    \end{bmatrix}
    \end{equation}
    [/tex]
    => rank(M)=2
     
  4. Feb 11, 2013 #3
    Re: The rank of a block matrix as a function of the rank of its submat

    Hi mfb,

    Thank you for answering! :)

    True! it depends on something more!

    M is also a covariance matrix so C is related to S1 and S2.

    It is a good idea to make the rank M dependent of the C rank.

    The rank of M may be dependent of the eigen values that are shared by S1 and S2

    Thankk you again

    All the best

    GoodSpirit
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: The rank of a block matrix as a function of the rank of its submatrice
  1. Rank of a matrix (Replies: 5)

  2. Rank of matrix (Replies: 2)

  3. Rank of a matrix (Replies: 3)

  4. Rank of a matrix (Replies: 1)

Loading...