Matrix norms

Matrix norms quantify matrix size; induced norms measure maximum vector amplification.

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Condition numbers
A matrix condition number describes sensitivity of linear-system solutions to perturbations.
Positive-definite matrices
A symmetric or Hermitian matrix is positive definite when its quadratic form is positive on nonzero vectors.
Basis and dimension
A basis is a linearly independent spanning set; every vector has unique coordinates in that basis.
Determinants
The determinant of a square matrix tracks signed volume scaling and is nonzero exactly when the matrix is invertible.
Diagonalization
A matrix is diagonalizable when it has a basis of eigenvectors; having repeated eigenvalues does not by itself decide diagonalizability.