# Linear Algebra II: Matrix Theory

### Course Description

This set contains linear algebra over fields other than R and topics concerning matrices, such as canonical forms and groups.

Lecture 9: Positive Semi-Definite Matrix 2: Spectral Theorem
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### Video Lectures & Study Materials

# Lecture Play Lecture
1 Matrix Inverse over the Complex Numbers (2:43) Play Video
2 Cramer's Rule over the Complex Numbers (4:13) Play Video
3 Gaussian Elimination over Z/3 (2:13) Play Video
4 Matrix Inverse over Z/7 (3:13) Play Video
5 Cramer's Rule over Z/5 (3:37) Play Video
6 Example of Skew-Symmetric Matrix (7:57) Play Video
7 Example of Simultaneous Diagonalization (7:10) Play Video
8 Positive Semi-Definite Matrix 1: Square Root (11:48) Play Video
9 Positive Semi-Definite Matrix 2: Spectral Theorem (12:39) Play Video
10 Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices (9:47) Play Video
11 Cayley-Hamilton Theorem for 2x2 Matrices (5:07) Play Video
12 Inverse of a Matrix Using the Cayley-Hamilton Theorem (2:54) Play Video
13 Cayley-Hamilton Theorem: General Case (16:23) Play Video
14 Cayley-Hamilton Theorem: Example 1 (4:08) Play Video
15 Cayley-Hamilton Theorem Example 2 (3:58) Play Video
16 Example of Invariant Subspace (10:41) Play Video
17 Overview of Jordan Canonical Form (15:54) Play Video
18 Example of Jordan Canonical Form: 2x2 Matrix (10:20) Play Video
19 Example of Jordan Canonical Form: General Properties (5:42) Play Video
20 Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1 (5:24) Play Video
21 Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2 (7:14) Play Video
22 Commutant of Complex Matrix (9:42) Play Video
23 Example of Rational Canonical Form 1: Single Block (17:33) Play Video
24 Example of Rational Canonical Form 2: Several Blocks (6:51) Play Video
25 Example of Rational Canonical Form 3 (19:10) Play Video
26 Exponential of 2x2 Matrix 1: Complex Case (6:56) Play Video
27 Exponential of 2x2 Matrix 2: Traceless Case (5:08) Play Video
28 Example of Group Action (8:34) Play Video
29 Example of Quaternions (5:35) Play Video
30 The Fibonacci Numbers Using Linear Algebra (HD Version) (12:25) Play Video
31 The Fibonacci Numbers Using Power Series (8:00) Play Video
32 Example of Group Automorphism 1 (Requires Linear Algebra) (17:36) Play Video
33 Example of Group Automorphism 2: G = Z/4 x Z/4 (Requires Linear Algebra) (11:24) Play Video
34 Group Theory: The Simple Group of Order 168 - Part 1 (10:43) Play Video
35 Group Theory: The Simple Group of Order 168 - Part 2 (14:47) Play Video