Matrix Properties

Common Matrix Properties

Property	Description
Invertibility	A matrix is invertible if there exists a two-sided inverse $A^{-1}$ .
Determinant	A scalar value summarizing a matrix's scaling and orientation properties.
Rank	Dimension of the column (or row) space of the matrix.
Trace	The sum of diagonal entries of a matrix.
Eigenvalues	Scalars $\lambda$ such that $Av = \lambda v$ for some vector $v$ .
Singular Values	Square roots of eigenvalues of $A^*A$ , used in SVD.
Norm	A measure of the matrix’s “size” or operator strength.
Condition Number	Describes sensitivity of output to input changes; related to stability.
Spectral Radius	The largest absolute value among eigenvalues.
Nullity	Dimension of the null space (kernel) of the matrix.
Orthogonality	A matrix has orthogonal rows or columns when their inner product is zero.
Linearity	All matrices represent linear maps between vector spaces.
Symmetry / Hermitian	A matrix equals its transpose (or conjugate transpose in complex case).
Positive Definiteness	For all nonzero vectors $x$ , $x^T A x > 0$ .
Idempotent	A matrix is idempotent if $A^2 = A$ .
Nilpotent	A matrix is nilpotent if $A^k = 0$ for some integer $k > 0$ .
Diagonalizability	A matrix is similar to a diagonal matrix $A = PDP^{-1}$ .
Jordan Form	A canonical form describing generalized eigenstructure.