An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 85
... tion 2.5.1 , p . 62 ) satisfies the identity ( x , y ) = = xTy = y1x . † = 0 for all en , show that EXERCISE 3.3.8 . Let A be an nxn matrix such that Ax vectors x ( of order n ) . By taking , in turn , x = A = 0 . e1 , ... , x ...
... tion 2.5.1 , p . 62 ) satisfies the identity ( x , y ) = = xTy = y1x . † = 0 for all en , show that EXERCISE 3.3.8 . Let A be an nxn matrix such that Ax vectors x ( of order n ) . By taking , in turn , x = A = 0 . e1 , ... , x ...
Page 385
... tion ( i.e. a combination of translations , rotations , and reflections ) can be shown to be a transformation of the type хо xo where = x0 , xx = Pkoo + Pk1x1 + Pk2x2 + Pk3 X3 ( k = 1,2,3 ) , P11 P12 P13 P21 P22 P23 P31 P32 P33 / is an ...
... tion ( i.e. a combination of translations , rotations , and reflections ) can be shown to be a transformation of the type хо xo where = x0 , xx = Pkoo + Pk1x1 + Pk2x2 + Pk3 X3 ( k = 1,2,3 ) , P11 P12 P13 P21 P22 P23 P31 P32 P33 / is an ...
Page 438
... tion of quadratic forms , 362-3 ; reduction of symmetric matrices , 302-4 ; set , 65–67 ; similarity , 266 ; vectors , 64 . Orthogonal matrices , 222-9 ; angle , 236 ; principal vector , 237 ; proper and improper , 233 ...
... tion of quadratic forms , 362-3 ; reduction of symmetric matrices , 302-4 ; set , 65–67 ; similarity , 266 ; vectors , 64 . Orthogonal matrices , 222-9 ; angle , 236 ; principal vector , 237 ; proper and improper , 233 ...
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