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Front matter from lecture notes by K. R. Matthews for course MP274 at the University of Queensland, 1991, typeset in LaTeX by Chris Fama. The contents list eight chapters: linear transformations, polynomials over a field, invariant subspaces, Jordan canonical form, rational canonical form, Smith canonical form, applications, and further directions. This is a third-party text kept in Phil's linear algebra folder, not his own work.
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LINEAR ALGEBRA NOTES
MP274 1991
K. R. MATTHEWS
LaTeXed by Chris Fama
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF QUEENSLAND
1991
Comments to the author at [email protected]
Contents
1 Linear Transformations 1
1.1 Rank + Nullity Theorems (for Linear Maps) . . . . . . . . . 3
1.2 Matrix of a Linear Transformation . . . . . . . . . . . . . . . 6
1.3 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Change of Basis Theorem for TA. . . . . . . . . . . . . . . . 18
2 Polynomials over a eld 20
2.1 Lagrange Interpolation Polynomials . . . . . . . . . . . . . . 21
2.2 Division of polynomials . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Euclid's Division Theorem . . . . . . . . . . . . . . . . 24
2.2.2 Euclid's Division Algorithm . . . . . . . . . . . . . . . 25
2.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . 26
2.4 Minimum Polynomial of a (Square) Matrix . . . . . . . . . . 32
2.5 Construction of a eld of pnelements . . . . . . . . . . . . . . 38
2.6 Characteristic and Minimum Polynomial of a Transformation 41
2.6.1Mnn(F[x])|Ring of Polynomial Matrices . . . . 42
2.6.2Mnn(F)[y]|Ring of Matrix Polynomials . . . . 43
3 Invariant subspaces 53
3.1T{cyclic subspaces . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 A nice proof of the Cayley-Hamilton theorem . . . . . 57
3.2 An Algorithm for Finding mT. . . . . . . . . . . . . . . . . . 58
3.3 Primary Decomposition Theorem . . . . . . . . . . . . . . . . 61
4 The Jordan Canonical Form 65
4.1 The Matthews' dot diagram . . . . . . . . . . . . . . . . . . . 66
4.2 Two Jordan Canonical Form Examples . . . . . . . . . . . . . 71
4.2.1 Example (a): . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Example (b): . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Uniqueness of the Jordan form . . . . . . . . . . . . . . . . . 75
4.4 Non{derogatory matrices and transformations . . . . . . . . . 78
4.5 Calculating Am, whereA2Mnn(C). . . . . . . . . . . . . . 79
4.6 Calculating eA, whereA2Mnn(C). . . . . . . . . . . . . . . 81
4.7 Properties of the exponential of a complex matrix . . . . . . . 82
4.8 Systems of di erential equations . . . . . . . . . . . . . . . . 87
4.9 Markov matrices . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.10 The Real Jordan Form . . . . . . . . . . . . . . . . . . . . . . 94
4.10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 94
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4.10.2 Determining the real Jordan form . . . . . . . . . . . 95
4.10.3 A real algorithm for nding the real Jordan form . . . 100
5 The Rational Canonical Form 105
5.1 Uniqueness of the Rational Canonical Form . . . . . . . . . . 110
5.2 Deductions from the Rational Canonical Form . . . . . . . . 111
5.3 Elementary divisors and invariant factors . . . . . . . . . . . 115
5.3.1 Elementary Divisors . . . . . . . . . . . . . . . . . . . 115
5.3.2 Invariant Factors . . . . . . . . . . . . . . . . . . . . . 116
6 The Smith Canonical Form 120
6.1 Equivalence of Polynomial Matrices . . . . . . . . . . . . . . 120
6.1.1 Determinantal Divisors . . . . . . . . . . . . . . . . . 121
6.2 Smith Canonical Form . . . . . . . . . . . . . . . . . . . . . . 122
6.2.1 Uniqueness of the Smith Canonical Form . . . . . . . 125
6.3 Invariant factors of a polynomial matrix . . . . . . . . . . . . 125
7 Various Applications of Rational Canonical Forms 131
7.1 An Application to commuting transformations . . . . . . . . 131
7.2 Tensor products and the Byrnes-Gauger theorem . . . . . . . 135
7.2.1 Properties of the tensor product of matrices . . . . . . 136
8 Further directions in linear algebra 143
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