Linear algebra notes

✍ Scribed by K.R. Matthews

• Publisher: Queensland
• Year: 1991
• Tongue: English
• Leaves: 146
• Category: Library

✦ Table of Contents

Title page
1 Linear Transformations
1.1 Rank + Nullity Theorems (for Linear Maps)
1.2 Matrix of a Linear Transformation
1.3 Isomorphisms
1.4 Change of Basis Theorem for TA
2 Polynomials over a field
2.1 Lagrange Interpolation Polynomials
2.2 Division of polynomials
2.2.1 Euclid's Division Theorem
2.2.2 Euclid's Division Algorithm
2.3 Irreducible Polynomials
2.4 Minimum Polynomial of a (Square) Matrix
2.5 Construction of a field of p^n elements
2.6 Characteristic and Minimum Polynomial of a Transformation
2.6.1 M_{nxn}(F[x])-Ring of Polynomial Matrices
2.6.2 M_{nxn}(F)[y]-Ring of Matrix Polynomials
3 Invariant subspaces
3.1 T-cyclic subspaces
3.1.1 A nice proof of the Cayley-Hamilton theorem
3.2 An Algorithm for Finding m_T
3.3 Primary Decomposition Theorem
4 The Jordan Canonical Form
4.1 The Matthews' dot diagram
4.2 Two Jordan Canonical Form Examples
4.2.1 Example (a):
4.2.2 Example (b):
4.3 Uniqueness of the Jordan form
4.4 Non-derogatory matrices and transformations
4.5 Calculating A^m, where A∈M_{nxn}(C)
4.6 Calculating e^A, where A∈M_{nxn(}C)
4.7 Properties of the exponential of a complex matrix
4.8 Systems of differential equations
4.9 Markov matrices
4.10 The Real Jordan Form
4.10.1 Motivation
4.10.2 Determining the real Jordan form
4.10.3 A real algorithm for finding the real Jordan form
5 The Rational Canonical Form
5.1 Uniqueness of the Rational Canonical Form
5.2 Deductions from the Rational Canonical Form
5.3 Elementary divisors and invariant factors
5.3.1 Elementary Divisors
5.3.2 Invariant Factors
6 The Smith Canonical Form
6.1 Equivalence of Polynomial Matrices
6.1.1 Determinantal Divisors
6.2 Smith Canonical Form
6.2.1 Uniqueness of the Smith Canonical Form
6.3 Invariant factors of a polynomial matrix
7 Various Applications of Rational Canonical Forms
7.1 An Application to commuting transformations
7.2 Tensor products and the Byrnes-Gauger theorem
7.2.1 Properties of the tensor product of matrices
8 Further directions in linear algebra