I have the following problem:
I need to implement a program that doesn't accept the matrix quadratic form $B^T\times B$ but it accepts the scalar quadratic form instead. Actually I need to find a simple way to write $trace((B^T\times B)^{-1})$ in scalar quadratic form where $B\in M_{(n,m)}$.
I know that the scalar quadratic form of $trace(B^T\times B)=\sum_{j=1}^mb_j^T\times b_j$ where $b_j$ is the $j^{th}$ column of $B$.
for instance if we have $trace([\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array}] \times [\begin{array}{cc} 1 & 3 \\ 2 & 4 \\ \end{array}])= [\begin{array}{cc} 1 & 2 \\ \end{array}]\times [\begin{array}{c} 1 \\ 2 \\ \end{array}]+ [\begin{array}{cc} 3 & 4 \\ \end{array}]\times [\begin{array}{c} 3 \\ 4 \\ \end{array}]=30$
And so I wonder if there is a similar representation for $trace((B^T\times B)^{-1})$.
If not I'm interested in finding a matrix factorization for a symmetric positive definite matrix that gives me $S=B^T\times B$. In other words, if we have $S$ we can find $B$ and vice versa. But here I'm excluding the Cholesky matrix decomposition since it deals only with triangular matrices.
Many thanks!