# Questions tagged [positive-definite]

For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.

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### Relating two seemingly different definitions of Riemannian metric.

I've been studying Riemannian geometry from John M. Lee's book on the same. My aim is to understand the geometry of Hermitian positive semidefinite matrices when viewed as a Riemannian manifold. In ...
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### Existence of a linear map to the space of SPD matrices

Does there exist a linear map from the set of 24x29 Matrices to the set of Symmetric Positive definite (SPD) 24x24 Matrices? I understand that matrix multiplication is a linear map but from the matrix ...
### Prove that $\text{tr}(B_1^{-1} B_2) \geq \text{tr}((A^\text{T} B_1 A)^{+} A^\text{T} B_2 A)$
Suppose that we have two real and positive definite $n \times n$ matrices $B_1$ and $B_2$ and that $A$ is an arbitrary real $n \times n$ matrix. Running some numerical tests by generating random ...