• • • • • • • • • • • • • •
properties of unitary matrices. two unitarily equivalent matrices have the same Frobenius norm. Schur’s theorem. Cayley-Hamilton theorem. eigenvalues of Hermiatian matrix are real. eigenvalues of PD matrix are positive. diagonalizability of Hermitian matrix. spectral representation for Hermitian matrix. SVD A is unitarily diagonalizable iff A is normal. QR factorization. SVD representation for Pseudo-inverse. Solutions for linear system . Every matrix is similar to a unique Jordan canonical form.
principles:
• • • • • • • • • • A span is a subspace. Each vector has a unique basis-representation. Cauchy-Schwarz inequality. Equivalence of norms. Every matrix has a symmetric part and a skew-symmetric part. Ax=b have a solution iff b belongs to the range (column space) of A. dim[null(A)]=n–rank(A), where A is an n by n matrix AB=I, then BA=I. Every basis can be orthonormalized (Gram-Schmidt orthonormalization). detA=∏detAii, where A is a quasi-triangular matrix.
concepts:
(lecture 1~2) subspace, span, basis, dimension, inner product, vector norm, vector norm deduced by inner product, adjoint, column space, range, RREF, rank, null space, nonsingular, orthogonal, orthonormal, quasitriangular (block triangular), quasi-diagonal (block diagonal), basis representation of linear transformation. (lecture 3) Matrix norm, operator norm (matrix norm deduced by the vector norm), Frobenius norm, maximum column sum norm , maximum row sum norm, spectral norm, eigenvalue-eigenvector equation, spectral radius, characteristic polynomial, algebraic multiplicity, principle minor, elementary systematic functions, similarity, diagonalizable, eigenspace, geometric multiplicity, nondefective. (lecture 4~6) unitary matrix, isometry, unitarily equivalent, Hermitian matrix, skew-Hermitian matrix, positive definite matrix, singular values, normal matrix. (lecture 7~8) generalized inverse, minimal polynomial, invariant subspace, generalized eigenspace, Jordan block, Jordan canonical form
n n
•
• • •ຫໍສະໝຸດ p ( A) x = p(λ ) x, if Ax = λx, x ≠ 0. trA = ∑ λi
i =1
and
det( A) = ∏ λi .
i =1
.
Some properties of similarity. diagonalizable conditions. Relationship of spectral radius and matrix norm.