An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 183
... SYMMETRIC if a¿¡ aji ( i , j = 1 , ... , n ) , i.e. if AT = A. † = The symmetry mentioned in the definition is , of course , symmetry with respect to the diagonal . EXERCISE 6.4.2 . If A is a rectangular matrix , show that ATA is a ...
... SYMMETRIC if a¿¡ aji ( i , j = 1 , ... , n ) , i.e. if AT = A. † = The symmetry mentioned in the definition is , of course , symmetry with respect to the diagonal . EXERCISE 6.4.2 . If A is a rectangular matrix , show that ATA is a ...
Page 301
... symmetric ) matrix with distinct characteristic roots is unitarily ( orthogonally ) similar to a diagonal matrix . Let A , ... , be the ( distinct ) characteristic roots of a hermitian or real symmetric matrix A , and denote by x1 ...
... symmetric ) matrix with distinct characteristic roots is unitarily ( orthogonally ) similar to a diagonal matrix . Let A , ... , be the ( distinct ) characteristic roots of a hermitian or real symmetric matrix A , and denote by x1 ...
Page 357
... symmetric . The insistence on symmetry enables us to set up a biunique correspondence between quad- ratic operators , quadratic forms ( defined below ) , and symmetric matrices . DEFINITION 12.1.8 . Any polynomial n $ ( X1 , ... , X12 ) ...
... symmetric . The insistence on symmetry enables us to set up a biunique correspondence between quad- ratic operators , quadratic forms ( defined below ) , and symmetric matrices . DEFINITION 12.1.8 . Any polynomial n $ ( X1 , ... , X12 ) ...
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