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Homework Help: Expressing a matrice as a sum of two non singular matrices

  1. Feb 22, 2012 #1
    Hello everyone , So here is this problem which i was recently thinking about
    Expressing any matrix as the sum of two non singular matrices
    So, when i think of ways to express a matrix as sum of two matrices, the thought which
    comes first is :

    (a) Any matrix can be expressed as the sum of a symmetric and an asymmetric matrix
    but an asymmetric matrix is always singular which means this option is ruled out

    (b) Suppose A and B are two non Singular matrices. There ought to be some technique
    of factorising A and B so that some common terms exist and when i combine these two, a term is obtained whose property defines whether the resulting matrix is singular or not

    What can be such a factorisation ?
    Last edited: Feb 22, 2012
  2. jcsd
  3. Feb 22, 2012 #2
    Perhaps express it as a sum of an upper-triangular and a lower-triangular matrix??

    When is a triangular matrix non-singular??
  4. Feb 22, 2012 #3


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    Science Advisor

    Where did you get that idea? First, "asymmetric" simply means "not symmetric"- you mean "anti-symmetric". And even for anti-symmetric matrices, this is not true:
    [tex]\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}[/tex]
    is anti-symmetric but has determinant 1 and so is not singular.
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