An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 122
... invariant ' - an ' invariant ' property being one that is shared by all matrices representing ( with respect to suitable bases ) the same linear operator as the given matrix . In view of Theorem 4.2.5 this simply means that a property ...
... invariant ' - an ' invariant ' property being one that is shared by all matrices representing ( with respect to suitable bases ) the same linear operator as the given matrix . In view of Theorem 4.2.5 this simply means that a property ...
Page 277
... INVARIANT SPACE of L. † Thus a vector space U is said to be an invariant space of L if its image space , under L , coincides with the original space . Every transformation possesses at least one invariant space , namely , that which ...
... INVARIANT SPACE of L. † Thus a vector space U is said to be an invariant space of L if its image space , under L , coincides with the original space . Every transformation possesses at least one invariant space , namely , that which ...
Page 278
... invariant space of L. n = Write U = LVn . In view of our hypothesis we have LU = U , and so I is an invariant space of L. Moreover , if V is an invariant space of L , i.e. if LV = V , then LV = V. Hence , since Vc Vn , V = LV C L⭑Vn Lk ...
... invariant space of L. n = Write U = LVn . In view of our hypothesis we have LU = U , and so I is an invariant space of L. Moreover , if V is an invariant space of L , i.e. if LV = V , then LV = V. Hence , since Vc Vn , V = LV C L⭑Vn Lk ...
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