An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 183
... matrices . DEFINITION 6.4.2 . The square matrix A = ( a ) of order n is 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 ...
... matrices . DEFINITION 6.4.2 . The square matrix A = ( a ) of order n is 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 ...
Page 303
... matrix B = S - 1AS thus has the form λι On - 1 B = B1 n - 1 where B1 is a real symmetric matrix of order n - 1 . Now , by the induction hypothesis , there exist an orthogonal matrix C1 and a diagonal matrix A1 , both of order n - 1 ...
... matrix B = S - 1AS thus has the form λι On - 1 B = B1 n - 1 where B1 is a real symmetric matrix of order n - 1 . Now , by the induction hypothesis , there exist an orthogonal matrix C1 and a diagonal matrix A1 , both of order n - 1 ...
Page 439
... matrix , 136-40 ; properties , 158–63 , 169 ; of a quadratic form , 360 ... symmetric matrices , 302-4 ; of quadratic forms , 369-74 ; of quadrics to ... symmetric matrices , 243-5 . Rigid motion , 246-7 . Rotation : axis , 240 ...
... matrix , 136-40 ; properties , 158–63 , 169 ; of a quadratic form , 360 ... symmetric matrices , 302-4 ; of quadratic forms , 369-74 ; of quadrics to ... symmetric matrices , 243-5 . Rigid motion , 246-7 . Rotation : axis , 240 ...
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