Symmetric Positive Definite Matrices

Symmetric Positive Definite Matrices or SPD matrices.

Definition: a square real matrix $A$ such that

1. $A^T=A$ and
2. $x^{T}Ax>0$ for any $x\ne 0$.

Eigendecomposition: $A=Q\Lambda Q^T$ with $Q$ real orthogonal and $\Lambda$ real.

SPD implies further that ${\lambda }_{ii}>0$.

Eigenvalues, Diagonalizable matrices, Unitarily diagonalizable matrices, Hermitian and symmetric matrices