Proof of non-singularity of triple matrix product

1. Aug 25, 2012

IniquiTrance

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

This is what I have:

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

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

Does this seem right?