An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 365
... principal axes . This conclusion is confirmed by the equation ( 12.2.1 ) which shows that Q is an ellipsoid or a hyperboloid , but not a quadric of revolution . Case II . ( Precisely two of A1 , A2 , λg are equal . ) = To fix our ideas ...
... principal axes . This conclusion is confirmed by the equation ( 12.2.1 ) which shows that Q is an ellipsoid or a hyperboloid , but not a quadric of revolution . Case II . ( Precisely two of A1 , A2 , λg are equal . ) = To fix our ideas ...
Page 367
... principal axes of Q. Thus each principal axis of Q is specified by a characteristic vector of A. 12.3 . General reduction to diagonal form In the preceding section we studied the reduction of quadratic forms to diagonal form by means of ...
... principal axes of Q. Thus each principal axis of Q is specified by a characteristic vector of A. 12.3 . General reduction to diagonal form In the preceding section we studied the reduction of quadratic forms to diagonal form by means of ...
Page 407
... principal minor , and by hypothesis all its negative principal minors are of odd order . Now if all principal minors of odd order were non- positive , then A would be negative definite or negative semi - definite Hence at least one ...
... principal minor , and by hypothesis all its negative principal minors are of odd order . Now if all principal minors of odd order were non- positive , then A would be negative definite or negative semi - definite Hence at least one ...
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