Linear algebra

Contents 1 Review of Matrix Algebra 2 1.1 Determinants, Minors, and Cofactors . . . . . . . . . . . . . . . . 4 1.2 Rank, Trace, and Inverse . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Elementary Operations and Matrices . . . . . . . . . . . . . . . . 7 2 Vectors and Linear (Vector) Spaces 9 2.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Eigenvalues and Eigenvectors 12 3.1 Jordan Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Inner Products & Norms 20 4.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Applications of Linear Algebra 24 5.1 The Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . 24 5.2 Solving State Equations . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Matrix Decompositions . . . . . . . . . . . . . . . . . . . . . . . 29 5.4 Bilinear Forms and Sign-definite Matrices . . . . . . . . . . . . . 32 5.4.1 Definite Matrices . . . . . . . . . . . . . . . . . . . . . . 32 5.5 Matrix Inversion Lemmas . . . . . . . . . . . . . . . . . . . . . . 33 6 MATLAB Commands 34

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