## Lambda-matrices and Vibrating SystemsFeatures aspects and solutions of problems of linear vibrating systems with a finite number of degrees of freedom. Starts with development of necessary tools in matrix theory, followed by numerical procedures for relevant matrix formulations and relevant theory of differential equations. Minimum of mathematical abstraction; assumes a familiarity with matrix theory, elementary calculus. 1966 edition. |

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### Contents

A SKETCH OF SOME MATRIX THEORY | 1 |

REGULAR PENCILS OF MATRICES AND EIGENVALUE PROBLEMS | 23 |

LAMBDAMATRICES I | 42 |

LAMBDAMATRICES II | 56 |

SOME NUMERICAL METHODS FOR LAMBDAMATRICES | 75 |

ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS | 100 |

THE THEORY OF VIBRATING SYSTEMS | 116 |

ON THE THEORY OF RESONANCE TESTING | 143 |

FURTHER RESULTS FOR SYSTEMS WITH DAMPING | 158 |

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### Common terms and phrases

A-matrix algorithm analysis applied arbitrary assume biorthogonal Chapter coefficients coincide columns compute consider constant corresponding damped system deduce defined denote derivatives DI(A diagonal matrix differential equations Dt(A eigen eigenvalues elementary divisors eqns exist finite functions gives hence hysteretic implies inverse iteration LAMBDA-MATRICES latent roots left eigenvectors left latent vectors lemma linear combination linearly independent linearly independent right method multiplicity n x n matrix natural frequencies non-negative definite non-singular matrix non-zero null eigenvalue obtain orthogonal partitions pencil of matrices perturbation positive definite premultiply problem proof proper numbers proper vectors properties proved q'Aq rate of convergence Rayleigh quotient real symmetric regular pencil right and left right eigenvectors right latent vectors roots and vectors satisfied scalar simple matrix pencil simple pencil simple structure solutions of 6.2.3 square matrix subspace of right symmetric matrix Theorem 2.5 theory transformation undamped natural frequencies vector space vectors q write zero