## A survey of matrix theory and matrix inequalities |

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### Contents

SURVEY OF MATRIX THEORY | 1 |

NUMBERS ASSOCIATED WITH MATRICES | 9 |

LINEAR EQUATIONS AND CANONICAL FORMS | 30 |

Copyright | |

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### Common terms and phrases

called charac characteristic polynomial characteristic roots characteristic roots Xi column operations commutes complex numbers convex combination convex function coordinates corresponding defined denote determinantal divisors diag diagonal matrix direct sum doubly stochastic matrices elementary divisors elementary matrices equal equivalent example exists a unitary field of values follows Frobenius fully indecomposable Hadamard Hence hermitian matrix indecomposable indecomposable matrix inequality inner product invariant factors LeVy-Desplanques theorem Linear algebra linearly independent main diagonal Marcus Math maximal root minimal polynomial Mn(C Moreover multiplicity n-square matrix n-tuples nonnegative hermitian nonnegative matrix nonsingular normal form normal matrix notation null space obtain orthonormal set permutation matrix positive definite hermitian positive integer positive semidefinite Proof prove real numbers result row and column row stochastic row sums sequences similarity invariants skew-hermitian skew-symmetric subdeterminants subset Suppose symmetric function teristic roots theorem unitary matrix vector space