Explains the spectral theorem for real symmetric matrices: their eigenvalues are real and their eigenvectors can be chosen orthonormal, giving the factorization A = QΛQᵀ. Also introduces positive definiteness.
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MIT 18.06: Linear Algebra
All
01-Geometry of Linear Equations
- 01-Geometry of Linear Equations
- 02-Elimination with Matrices
- 03-Multiplication and Inverse Matrices
- 04-Factorization into A = LU
- 05-Transposes, Permutations, Spaces R^n
- 06-Column Space and Nullspace
- 07-Solving Ax = 0: Pivot Variables, Special Solutions
- 08-Solving Ax = b: Row Reduced Form R
- 09-Independence, Basis, and Dimension
- 10-Four Fundamental Subspaces
- 11-Matrix Spaces; Rank 1; Small World Graphs
- 12-Graphs, Networks, Incidence Matrices
- 13-Quiz 1 Review
- 14-Orthogonal Vectors and Subspaces
- 15-Projections onto Subspaces
- 16-Projection Matrices and Least Squares
- 17-Orthogonal Matrices and Gram-Schmidt
- 18-Properties of Determinants
- 19-Determinant Formulas and Cofactors
- 20-Cramer's Rule, Inverse Matrix, and Volume
- 21-Eigenvalues and Eigenvectors
- 22-Diagonalization and Powers of A
- 23-Differential Equations and exp(At)
- 24-Markov Matrices; Fourier Series
- 25-Quiz 2 Review
- 26-Symmetric Matrices and Positive Definiteness
- 27-Complex Matrices; Fast Fourier Transform
- 28-Positive Definite Matrices and Minima
- 29-Similar Matrices and Jordan Form
- 30-Singular Value Decomposition
- 31-Linear Transformations and Their Matrices
- 32-Change of Basis; Image Compression
- 33-Quiz 3 Review
- 34-Left and Right Inverses; Pseudoinverse
- 35-Final Course Review