An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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... EXERCISE 2.2.2 . Show that , for all X , Y = M and all a , ẞ e F , a ( X - Y ) = αX - aY , ( α - B ) X = αX - BX . EXERCISE 2.2.3 . Show that , if a 0 , X 0 , then aX ‡ ☺ . k EXERCISE 2.2.4 . Let X1 , ... , X be generators of a linear ...
... EXERCISE 2.2.2 . Show that , for all X , Y = M and all a , ẞ e F , a ( X - Y ) = αX - aY , ( α - B ) X = αX - BX . EXERCISE 2.2.3 . Show that , if a 0 , X 0 , then aX ‡ ☺ . k EXERCISE 2.2.4 . Let X1 , ... , X be generators of a linear ...
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... EXERCISE 11.1.2 . Let A be a complex matrix , B a non - negative matrix , and m a positive integer . Show that Bm is non - negative and that , if AB , then Am Bm . EXERCISE 11.1.3 . Let { Am } be a sequence of complex matrices , { B } a ...
... EXERCISE 11.1.2 . Let A be a complex matrix , B a non - negative matrix , and m a positive integer . Show that Bm is non - negative and that , if AB , then Am Bm . EXERCISE 11.1.3 . Let { Am } be a sequence of complex matrices , { B } a ...
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... Exercise 11.1.2 , U - 1AmUTm = SAMS - 1 . Hence , by Exercises 11.1.4 and 11.1.5 , || U - 1A TM U || ≤ || SAmS - 1 || ≤ n2 || S || . || A TM || . || S - 1 || < n2 || S || . || S - 1 || . qm → 0 . Consequently , by Exercise 11.1.6 , U ...
... Exercise 11.1.2 , U - 1AmUTm = SAMS - 1 . Hence , by Exercises 11.1.4 and 11.1.5 , || U - 1A TM U || ≤ || SAmS - 1 || ≤ n2 || S || . || A TM || . || S - 1 || < n2 || S || . || S - 1 || . qm → 0 . Consequently , by Exercise 11.1.6 , U ...
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