Product of matrices: $\mathbf Q^{-1}\cdot\mathbf B^T\cdot\mathbf B\cdot\mathbf Q$ Let $\mathbf Q$ be an orthogonal matrix ($\mathbf Q^T\cdot\mathbf Q=\mathbf I$). If I have another matrix $\mathbf B$, is there any special propriety found in the following expression? $$ \mathbf Q^T\cdot\mathbf B^T\cdot\mathbf B\cdot\mathbf Q $$ Or equivalently: $$ \mathbf Q^{-1}\cdot\mathbf B^T\cdot\mathbf B\cdot\mathbf Q $$ How does it relate to $\mathbf B$? Thank you very much.

For any matrix $B$, the matrix $B^TB$ will be symmetric and positive semidefinite. This matrix is sometimes referred to as the "Grammian matrix" or the "covariance matrix" associated with $B$.

The matrix $Q^T(B^TB)Q$ will be symmetric and positive definite as well. It will also be similar to the matrix $B^TB$. Note that this matrix can also be written in the form $(BQ)^T(BQ)$.