# Finding a diagonal matrix for a given matrix

Let $A$ be a real $n \times n$ matrix. Then there exist orthogonal matrices $T_1$ and $T_2$ such that $A=T_1 D T_2$ where D is a diagonal matrix with positive entries. D is unique up permutation of the diagonal entries. This decomposition is sometimes called Cartan decomposition.

Question: Is it possible to obtain the matrix $D$ for a given matrix $A$ in Sage?

This might be of some interest for you... as well as this answer.

Not sure but have a look at A.eigenmatrix_left() and eigenmatrix_right() , the eigenvectors are orthogonal and the eigenmatrix is diagonal.