矩陣理論學習導引

以下是 2010 年秋我在交大電機系開授「矩陣理論」所準備的參考連結,之後並陸續添增新連結。「矩陣理論」是基礎線性代數的延伸進階課程,授課對象是本系三四年級學生。

 
Part I. Vector Spaces and Linear Transformations

Lecture 1 Vector Spaces
Lecture 2 Basis and Dimension
Lecture 3 Linear Transformations
Lecture 4 Rank-Nullity Theorem
Lecture 5 Isomorphism
Lecture 6 Matrices of Transformations
Lecture 7 Matrix Algebra
Lecture 8 Elementary Row Operations
Lecture 9 Block Matrices
Lecture 10 Four Fundamental Subspaces of Matrices
Lecture 11 Calculus of Subspaces
Lecture 12 Rank of AB
Lecture 13 Invariant Subspaces
Lecture 14 Change of Basis/Coordinates

 
Part II. Eigenvalues and Eigenvectors

Lecture 15 Determinants
Lecture 16 Course Review I
Lecture 17 Cramer’s Rule and Adjugate
Lecture 18 The Eigenvalue-Eigenvector Equation
Lecture 19 Diagonalization
Lecture 20 Multiplicity
Lecture 21 Similarity
Lecture 22 Characterization of Similar Matrices
Lecture 23 Differential Equations and Matrix Exponential

 
Part III. Orthogonality

Lecture 24 Inner Product Spaces
Lecture 25 Orthgonal Complement and Orthogonal Projection
Lecture 26 Gram-Schmidt Process and QR factorization
Lecture 27 Least-Squares Method and Orthogonal/Unitary Matrices

 
Part IV. Jordan Normal Form

Lecture 28 Schur Decomposition
Lecture 29 Normal Matrices
Lecture 30 Jordan Normal Form (I)
Lecture 31 Jordan Normal Form (II)
Lecture 32 Course Review II
Lecture 33 Applications of Jordan Form
Lecture 34 Cayley-Hamilton Theorem
Lecture 35 The Minimal Polynomial

 
Part V. Quadratic Forms

Lecture 36 Hermitian Matrices
Lecture 37 Positive Definite Matrices
Lecture 38 Skew-Symmetric Matrices
Lecture 39 Rayleigh Quotient
Lecture 40 Singular Values
Lecture 41 Singular Value Decomposition
Lecture 42 Pseudoinverses
Lecture 43 Matrix Norms

Lecture 44 Course Review III

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