An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 354
... FORM in the two sets of variables x1 , ... , Xm The m × n matrix A is called the matrix of this bilinear form . and Y1 , ... , Yn There is thus clearly a biunique correspondence ... BILINEAR , QUADRATIC , AND HERMITIAN FORMS XII , § 12.1.
... FORM in the two sets of variables x1 , ... , Xm The m × n matrix A is called the matrix of this bilinear form . and Y1 , ... , Yn There is thus clearly a biunique correspondence ... BILINEAR , QUADRATIC , AND HERMITIAN FORMS XII , § 12.1.
Page 361
Leonid Mirsky. quadratic form gives rise unambiguously to a symmetric bilinear form , i.e. to a bilinear form whose matrix is symmetric . The neces- sary process of symmetrization was , in effect , carried out on p . 357 , but the ...
Leonid Mirsky. quadratic form gives rise unambiguously to a symmetric bilinear form , i.e. to a bilinear form whose matrix is symmetric . The neces- sary process of symmetrization was , in effect , carried out on p . 357 , but the ...
Page 368
... bilinear form of rank r can be changed into the bilinear form X1Y1 + ... + X , Y , by means of a non - singular linear transformation . ( 12.3.1 ) This result states , in fact , that if is a bilinear operator , of rank r , on the linear ...
... bilinear form of rank r can be changed into the bilinear form X1Y1 + ... + X , Y , by means of a non - singular linear transformation . ( 12.3.1 ) This result states , in fact , that if is a bilinear operator , of rank r , on the linear ...
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