Linear Algebra (full)

A\mathbf{x} = \lambda \mathbf{x}

A = Q \Lambda Q^{-1}, \quad \Lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)

A = U \Sigma V^{\top}, \quad U^{\top} U = V^{\top} V = I

P_{\!V} = V (V^{\top} V)^{-1} V^{\top}

\det(A) = \prod_{i=1}^{n} \lambda_i, \quad \operatorname{tr}(A) = \sum_{i=1}^{n} \lambda_i