Prove a matrix $A Q^{-1}A^T$ is invertible

Question

Let $A$ be $m \times n$ matrix with linearly independent rows and let $Q$ be a square matrix of type $n \times n$ and positive definite. Assume that $Q$ is invertible. Show that $A Q^{-1} A^T$ is invertible.

I know that if the rows of $A$ are linearly independent so that $AA^T$, which is $m \times m$ matrix, is invertible. Proof can be found in Prove

But in this case, I don't have any idea to begin.

I remember working with this problem. I think rank of $AQ^{-1}$A' is equal to rank of $AA'$, thus it is invertible — Lee, May 31 '21 at 03:36
@Lee but how to prove rank of $AQ^{-1}A^T$ is equal to rank of $AA^T$? — FactorY, May 31 '21 at 03:45
closely related: https://math.stackexchange.com/questions/3724212/how-big-can-the-null-space-of-atsa-be/ — user8675309, May 31 '21 at 18:19

score 4 · Accepted Answer · answered May 31 '21 at 05:00

4

You need further hypotheses (e.g. $Q$ being positive definite). The claim is wrong as stated. Example: $A=\pmatrix{ 1 & 1}$ and $Q=\pmatrix{1 & 0 \\ 0 & -1}$ yields $A Q^{-1}A^T=\pmatrix{0}$

answered May 31 '21 at 05:00

H. H. Rugh

35,236

Prove a matrix $A Q^{-1}A^T$ is invertible

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