>[!success] Decomposition of psd Matrices >A $n\times n$ [[Matrices|matrix]] $\mathbf{A}$ is positive semi-definite if and only if it can be decomposed into a product $\mathbf{A}=\mathbf{B}^*\mathbf{B},$where again the conjugate is replaced by a transpose in the real case $\mathbf{A}=\mathbf{B}^\top \mathbf{B}$. $\mathbf{A}$ is positive definit if and only if $\mathbf{B}$ is invertible (full rank). For any decomposition, it holds that $\text{rank}(\mathbf{A})=\text{rank}(\mathbf{B}).$