An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 215
... least an 8 - fold charac- teristic root of wI - A , i.e. w is at least an s - fold characteristic root of A. Hence k 8 , and this implies ( 7.6.1 ) . It should be noted that strict inequality in ( 7.6.1 ) may actually occur . Thus ...
... least an 8 - fold charac- teristic root of wI - A , i.e. w is at least an s - fold characteristic root of A. Hence k 8 , and this implies ( 7.6.1 ) . It should be noted that strict inequality in ( 7.6.1 ) may actually occur . Thus ...
Page 391
... least value assumed by the quadratic form XTATAX , subject to the condition xx = 1 , is ( 9 - √65 ) / 2 . 13. Let à be a characteristic root of the complex matrix A , and let M and m be the greatest and least characteristic roots of ...
... least value assumed by the quadratic form XTATAX , subject to the condition xx = 1 , is ( 9 - √65 ) / 2 . 13. Let à be a characteristic root of the complex matrix A , and let M and m be the greatest and least characteristic roots of ...
Page 397
... least one root is equal to zero . ( iii ) is indefinite if and only if A has at least one positive and at least one negative characteristic root . When A is real and symmetric this result enables us ( in view of Theorem 13.1.1 ) to ...
... least one root is equal to zero . ( iii ) is indefinite if and only if A has at least one positive and at least one negative characteristic root . When A is real and symmetric this result enables us ( in view of Theorem 13.1.1 ) to ...
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