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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
    2017 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
     
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