# Enforcing sum of Hermitian matrices to be positive definite

I have a set of Hermitian matrices $A_i$ which are orthonormal in the Frobenius inner product $\langle A_i,A_j\rangle=\mathrm{Tr}(A_i^{\dagger}A_j)=\delta_{ij}$, and I know there exists a basis in which the orthonormality and Hermiticity are preserved but all matrices are positive definite. Expressing the matrices in the positive definite basis as

$$B_i=\sum_{\gamma} c_{\gamma i}A_{\gamma},$$

the Hermiticity and orthonormality requirements require that $c$ be an orthogonal matrix. Is there a simple means of finding the transformation matrix $c$?

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