An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 28
... assume , without loss of generality , that a110 . In that case we subtract , for 2 , ... , n , ai / a11 times the first row from the ith row in D and 122 i = obtain ban a11 a12 ain 0 b22 ban = a11 D = bnz bnn • 0 bn2 bnn a1a1j where bij ...
... assume , without loss of generality , that a110 . In that case we subtract , for 2 , ... , n , ai / a11 times the first row from the ith row in D and 122 i = obtain ban a11 a12 ain 0 b22 ban = a11 D = bnz bnn • 0 bn2 bnn a1a1j where bij ...
Page 67
... Assume now that , for some k > 2 , Y1 , ... , Yk - 1 are all non - zero . Then the definition of y , in ( 2.5.5 ) is significant and we recognize , further- more , that y is a linear combination of X ,, ... , X in which x has the ...
... Assume now that , for some k > 2 , Y1 , ... , Yk - 1 are all non - zero . Then the definition of y , in ( 2.5.5 ) is significant and we recognize , further- more , that y is a linear combination of X ,, ... , X in which x has the ...
Page 372
... Assume , then , without loss of generality , that a120 and use the transformation X2 -- $ 1 + $ 2 , xx = x ( k2 ) . n This transformation has determinant 1 and so is non - singular . It carries & into a quadratic form in §1 ...
... Assume , then , without loss of generality , that a120 and use the transformation X2 -- $ 1 + $ 2 , xx = x ( k2 ) . n This transformation has determinant 1 and so is non - singular . It carries & into a quadratic form in §1 ...
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