An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 150
... proof just given we speak of the matrix A rather than of a linear transformation , the argument is , in fact , of the ' invariant ' type ; and of this the reader should have no difficulty in satisfying himself . The ' invariant ...
... proof just given we speak of the matrix A rather than of a linear transformation , the argument is , in fact , of the ' invariant ' type ; and of this the reader should have no difficulty in satisfying himself . The ' invariant ...
Page 306
... proof is now completed by induction with respect to n . If the theorem holds for n - 1 , then there exists a unitary matrix C1 and a diagonal matrix A1 , both of order n - 1 , such that B1 C1 = С1 A1 . Hence 0 0 0 1 B ) ( c ) = ( c ) ...
... proof is now completed by induction with respect to n . If the theorem holds for n - 1 , then there exists a unitary matrix C1 and a diagonal matrix A1 , both of order n - 1 , such that B1 C1 = С1 A1 . Hence 0 0 0 1 B ) ( c ) = ( c ) ...
Page 413
... proof is now completed in precisely the same way as the proof of Theorem 13.4.2 . EXERCISE 13.4.3 . Supply the details of the proof given above in outline . The simultaneous transformation described by Theorem 13.4.7 is effected in a ...
... proof is now completed in precisely the same way as the proof of Theorem 13.4.2 . EXERCISE 13.4.3 . Supply the details of the proof given above in outline . The simultaneous transformation described by Theorem 13.4.7 is effected in a ...
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