Given $n \times (n+m-1)$ Toeplitz matrices $A$ and $B$, if $AB^T$ is positive definite, how to prove that
$$\left( A^T - B^T \right) \left( BB^T \right)^{-1} B + I$$
is also positive definite?
2026-03-25 13:32:50.1774445570
Is this positive definite?
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Convolution matrix is Toeplitz and upper triangular matrix with dimensions n$\times$(n+m-1):
\begin{pmatrix} a_1 & a_2 & a_3 & .. & a_m & 0 & ..& & 0\\ 0 & a_1 & a_2 & a_3 & .. & a_m & 0 & .. & 0\\ & .. & .. &.. & .. & .. & .. & .. \\ 0 & .. & 0 & 0 & 0 & .. & a_{m-1} &a_{m-1} & a_m & \end{pmatrix}
"*" means matrix multiplication. I corrected it.
Note that $(BB^T)^{-1}B=B^\dagger$, where $\dagger$ denotes pseudoinversion.