4.7 Matrix Phylogeny (Overview)

Matrices can be classified based on properties and decompositions:

Matrix type	Property
Square, invertible	Determinant \(\neq 0\)
Non-defective	Diagonalizable, has \(n\) independent eigenvectors
Normal	\(\mathbf{A}^\top \mathbf{A} = \mathbf{A}\mathbf{A}^\top\)
Orthogonal	\(\mathbf{A}^\top \mathbf{A} = \mathbf{A}\mathbf{A}^\top = I\), subset of invertible matrices
Symmetric	\(S = S^\top\), real eigenvalues
Positive definite	\(x^\top P \mathbf{x} > 0\) for all \(x \neq 0\), unique Cholesky decomposition
Diagonal	Closed under multiplication/addition, special case: identity matrix \(I\)

SVD exists for all real matrices, square or rectangular. Eigenvalue decomposition exists only for non-defective square matrices. The phylogenetic relationships between matrix types help organize matrix operations and decompositions.