Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices
by
Robert Donley

### Lecture Description

Matrix Theory: Let A be an invertible nxn matrix with complex entries. Using the square root result from Part 1, we show that A factors uniquely as PX, where P is unitary and X is (Hermitian) positive definite.

### Course Index

- Matrix Inverse over the Complex Numbers
- Cramer's Rule over the Complex Numbers
- Gaussian Elimination over Z/3
- Matrix Inverse over Z/7
- Cramer's Rule over Z/5
- Example of Skew-Symmetric Matrix
- Example of Simultaneous Diagonalization
- Positive Semi-Definite Matrix 1: Square Root
- Positive Semi-Definite Matrix 2: Spectral Theorem
- Positive Semi-Definite Matrix 3: Factorization of Invertible Matrices
- Cayley-Hamilton Theorem for 2x2 Matrices
- Inverse of a Matrix Using the Cayley-Hamilton Theorem
- Cayley-Hamilton Theorem: General Case
- Cayley-Hamilton Theorem: Example 1
- Cayley-Hamilton Theorem Example 2
- Example of Invariant Subspace
- Overview of Jordan Canonical Form
- Example of Jordan Canonical Form: 2x2 Matrix
- Example of Jordan Canonical Form: General Properties
- Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 1
- Example of Jordan Canonical Form: Real 4x4 Matrix with Basis 2
- Commutant of Complex Matrix
- Example of Rational Canonical Form 1: Single Block
- Example of Rational Canonical Form 2: Several Blocks
- Example of Rational Canonical Form 3
- Exponential of 2x2 Matrix 1: Complex Case
- Exponential of 2x2 Matrix 2: Traceless Case
- Example of Group Action
- Example of Quaternions
- The Fibonacci Numbers Using Linear Algebra (HD Version)
- The Fibonacci Numbers Using Power Series
- Example of Group Automorphism 1 (Requires Linear Algebra)
- Example of Group Automorphism 2: G = Z/4 x Z/4 (Requires Linear Algebra)
- Group Theory: The Simple Group of Order 168 - Part 1
- Group Theory: The Simple Group of Order 168 - Part 2

### Course Description

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

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