An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 96
... zero now naturally presents itself . We shall solve it here for the case of square matrices and deal with the general case in § 5.5.2 . THEOREM 3.6.8 . A ( non - zero ) square matrix is a divisor of zero if and only if it is singular ...
... zero now naturally presents itself . We shall solve it here for the case of square matrices and deal with the general case in § 5.5.2 . THEOREM 3.6.8 . A ( non - zero ) square matrix is a divisor of zero if and only if it is singular ...
Page 154
... zero ) non - square matrix is necessarily a divisor of zero . We recall that a matrix A ‡ O is called a divisor of zero if and only if there exists a matrix XO such that AX = O or a matrix YO such that YA O. In the former case X ...
... zero ) non - square matrix is necessarily a divisor of zero . We recall that a matrix A ‡ O is called a divisor of zero if and only if there exists a matrix XO such that AX = O or a matrix YO such that YA O. In the former case X ...
Page 203
... zero polynomial annihilating A is proved once again . Among all non - zero polynomials annihilating A we now consider those of least degree , and by multiplying them by suitable non - zero constants we ensure that they are monic ( i.e. ...
... zero polynomial annihilating A is proved once again . Among all non - zero polynomials annihilating A we now consider those of least degree , and by multiplying them by suitable non - zero constants we ensure that they are monic ( i.e. ...
Table des matières
PART | 1 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
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Expressions et termes fréquents
A₁ algebra assertion automorphism B₁ basis bilinear form bilinear operator canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix dimensionality E-operations equal equivalence EXERCISE exists follows functions given Hence hermitian form hermitian matrix identity implies inequality integers invariant space isomorphic linear equations linear manifold linear transformation linearly independent matrix group matrix of order minimum polynomial multiplication non-singular linear transformation non-singular matrix non-zero numbers nxn matrix obtain orthogonal matrix positive definite positive semi-definite possesses proof of Theorem prove quadratic form quadric rank relation represented respect result rotation S-¹AS satisfies scalar Show similar singular skew-symmetric matrix solution square matrix suppose symmetric matrix t₁ tion triangular unique unit element unitary matrix V₁ values vector space view of Theorem w₁ write x₁ xTAx y₁ zero