LINEAR ALGEBRA NOTES MP274 1991 - Number Theory Web

LINEAR ALGEBRA NOTES MP274 1991

K. R. MATTHEWS LaTeXed by Chris Fama

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF QUEENSLAND

1991

Comments to the author at krm@maths.uq.edu.au

Contents 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 . . . . . . .

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1 3 6 12 18

2 Polynomials over a field 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 field of pn elements . . . . . . . . . . . . . . 38 2.6 Characteristic and Minimum Polynomial of a Transformation 41 2.6.1 Mn×n (F [x])—Ring of Polynomial Matrices . . . . 42 2.6.2 Mn×n (F )[y]—Ring of Matrix Polynomials . . . . 43 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 mT . . . . . . . . . . . . . 3.3 Primary Decomposition Theorem . . . . . . . . . . .

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53 54 57 58 61

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 Am , where A ∈ Mn×n (C). . . . . . . . 4.6 Calculating eA , where A ∈ Mn×n (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 . . . . . . . . . . . . . . . . . .

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65 66 71 71 73 75 78 79 81 82 87 89 94 94

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4.10.2 Determining the real Jordan form . . . . . . . . . . . 95 4.10.3 A real algorithm for finding the real Jordan form . . . 100 5 The 5.1 5.2 5.3

Rational Canonical Form Uniqueness of the Rational Canonical Form . . Deductions from the Rational Canonical Form 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 6.3 Invariant factors of a polynomial matrix .

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. . . . . . . . . . . . Form . . . .

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105 . 110 . 111 . 115 . 115 . 116

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120 . 120 . 121 . 122 . 125 . 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

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