## Applications of the theory of matrices |

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### Contents

Chapter | 1 |

Some formulas for complex orthogonal and unitary matrices | 2 |

Polar decomposition of a complex matrix | 7 |

Copyright | |

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### Common terms and phrases

adjoint matrix antisymmetric matrix arbitrary block diagonal canonical form Cauchy index columns Conversely Corollary corresponding defined degree denote diagonal blocks differential equations dominant characteristic number dominant characteristic root elementary divisors elements equal to zero finite following theorem formula Frobenius given pencil Hankel matrix Hence Hermitian forms holds homogeneous Markov chain Hurwitz determinants Hurwitz polynomial inequalities integral irreducible matrix lemma limiting absolute probabilities limiting transition probabilities linear linearly independent minimal indices modulus multiplied necessary and sufficient nonnegative matrix nonsingular nonzero normal form obtain orthogonal matrix pencil of matrices permutation positive characteristic vector primitive matrix principal minors proof quadratic forms rank rational function real numbers reducible regular pencil relation replaced right member Routh scheme Routh-Hurwitz rows satisfy sequence stochastic matrix strictly equivalent sufficient condition symmetric matrix theorem is proved totally nonnegative XB and A1