An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Leonid Mirsky. Positive definite : x2 + y2 Positive semi - definite : ( x + y ) 2 Negative definite : -x2 - y2 Negative semi - definite : - ( x + y ) 2 Indefinite : x2 - y2 If is positive definite ( positive semi - definite ) , then - is ...
Leonid Mirsky. Positive definite : x2 + y2 Positive semi - definite : ( x + y ) 2 Negative definite : -x2 - y2 Negative semi - definite : - ( x + y ) 2 Indefinite : x2 - y2 If is positive definite ( positive semi - definite ) , then - is ...
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... definite form singular , and that an indefinite form can be either . Theorem 13.1.4 can easily be restated in terms of rank and signature . THEOREM ... DEFINITE AND INDEFINITE FORMS XIII , § 13.1 Transformations of positive definite forms.
... definite form singular , and that an indefinite form can be either . Theorem 13.1.4 can easily be restated in terms of rank and signature . THEOREM ... DEFINITE AND INDEFINITE FORMS XIII , § 13.1 Transformations of positive definite forms.
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Leonid Mirsky. 27. Let be a real , positive definite matrix , and let a real matrix A be called -orthogonal if the substitution x = Ay leaves the quadratic form xTx ... positive definite quadratic form XIII 425 PROBLEMS ON CHAPTER XIII.
Leonid Mirsky. 27. Let be a real , positive definite matrix , and let a real matrix A be called -orthogonal if the substitution x = Ay leaves the quadratic form xTx ... positive definite quadratic form XIII 425 PROBLEMS ON CHAPTER XIII.
Table des matières
PART | 1 |
VECTOR SPACES AND LINEAR MANIFOLDS | 39 |
THE ALGEBRA OF MATRICES | 72 |
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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