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Proof of non-singularity of triple matrix product

  1. Aug 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose [itex]A\in \mathbb{R}^{n\times n}[/itex] is symmetric positive definite, and therefore non-singular. Let [itex]M\in\mathbb{R}^{m\times n}[/itex]. Show that the matrix [itex]M^T A M[/itex] is non-singular if and only if the columns of [itex]M[/itex] are linearly independent.

    2. Relevant equations



    3. The attempt at a solution

    This is what I have:

    If the column vectors of [itex]M[/itex] are linearly independent, then since [itex]M^T AM= col(m_i^T A m_i)_{i=1:n}[/itex], and [itex]A[/itex] is SPD, the columns of [itex]M^T A M[/itex] are all independent non-zero vectors. Thus it is non-singular.

    In the other direction, if [itex]M^T A M[/itex] is non-singular, then it's column vectors must be linearly independent. Since its columns are all [itex]col(m_i^T A m_i)_{i=1:n}[/itex], the non-singularity of [itex]M^T A M[/itex] implies they are all linearly independent of each other, which can only be true if the [itex]m_i[/itex] are linearly independent of each other.

    Does this seem right?
     
  2. jcsd
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