An Introduction to Linear AlgebraClarendon Press, 1972 - 440 pages |
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Page 35
... Prove that E = — ( n − 2 ) 2n − 1D . n - 11. D is the n - rowed determinant in which the elements in the diagonal ... prove that , if a ( 0 < 0 < π ) , then Dn 13. Prove that = sin ( n + 1 ) 0 / sin 0 . = 1 , x = 2 cos 0 1 + α1 аг аз ...
... Prove that E = — ( n − 2 ) 2n − 1D . n - 11. D is the n - rowed determinant in which the elements in the diagonal ... prove that , if a ( 0 < 0 < π ) , then Dn 13. Prove that = sin ( n + 1 ) 0 / sin 0 . = 1 , x = 2 cos 0 1 + α1 аг аз ...
Page 166
... Prove that , if A is a G - matrix , then | A | : ± 1 . If A ( a ) and B ( bij ) are two G - matrices and if | A | + | B | = 0 , prove that the system of equations = = ( Aƒƒ Xj + bjj xj ) = 0 ( i 1 , ... , n ) possesses a non - trivial ...
... Prove that , if A is a G - matrix , then | A | : ± 1 . If A ( a ) and B ( bij ) are two G - matrices and if | A | + | B | = 0 , prove that the system of equations = = ( Aƒƒ Xj + bjj xj ) = 0 ( i 1 , ... , n ) possesses a non - trivial ...
Page 193
... prove that aii = r . i = 1 16. Use Theorem 6.2.4 ( p . 177 ) to prove Theorem 5.6.5 ( p . 162 ) for the case of square matrices . 17. By a modification of the proof of Theorem 6.4.1 ( p . 183 ) show that , if A is a matrix such that ĀT ...
... prove that aii = r . i = 1 16. Use Theorem 6.2.4 ( p . 177 ) to prove Theorem 5.6.5 ( p . 162 ) for the case of square matrices . 17. By a modification of the proof of Theorem 6.4.1 ( p . 183 ) show that , if A is a matrix such that ĀT ...
Table des matières
DETERMINANTS VECTORS MATRICES | 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 C₁ canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix 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 values vector space view of Theorem write x₁ xTAx y₁ zero α₁