An Introduction to Linear AlgebraClarendon Press, 1972 - 440 pages |
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Page 327
... Convergent matrix sequences 11.1.1 . An analytic theory of matrices must naturally be founded on the notion of convergence or some equivalent notion . DEFINITION 11.1.1 . The sequence { Am } of nxn matrices CON- VERGES ( or TENDS TO ) A ...
... Convergent matrix sequences 11.1.1 . An analytic theory of matrices must naturally be founded on the notion of convergence or some equivalent notion . DEFINITION 11.1.1 . The sequence { Am } of nxn matrices CON- VERGES ( or TENDS TO ) A ...
Page 331
... convergent matrix series , then m = 0 Σ PAQ is also convergent , and m = 0 ∞ Σ PAMQ = P m = 0 P ( Am ) Q . DEFINITION 11.2.2 . The series ( 11.2.1 ) is ABSOLUTELY CONVER- GENT if each of the series on the left - hand side of ( 11.2.2 ) ...
... convergent matrix series , then m = 0 Σ PAQ is also convergent , and m = 0 ∞ Σ PAMQ = P m = 0 P ( Am ) Q . DEFINITION 11.2.2 . The series ( 11.2.1 ) is ABSOLUTELY CONVER- GENT if each of the series on the left - hand side of ( 11.2.2 ) ...
Page 337
... convergent for | z | < R and therefore , by Theorem m = 0 ∞ 11.2.4 ( i ) , dm Am is absolutely convergent . Hence , by Theorem 11.2.1 , m = 0 ∞ m ( 2 | a , || bm - v | ) || Am || | ) || Am || = Ž || dm Am || m = 0v = 0 Σ m = 0 is ...
... convergent for | z | < R and therefore , by Theorem m = 0 ∞ 11.2.4 ( i ) , dm Am is absolutely convergent . Hence , by Theorem 11.2.1 , m = 0 ∞ m ( 2 | a , || bm - v | ) || Am || | ) || Am || = Ž || dm Am || m = 0v = 0 Σ m = 0 is ...
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