An Introduction to Linear AlgebraClarendon Press, 1955 - 433 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 276
... invariant methods ; that is , by making use only of the intrinsic properties of linear trans- formations and not of their matrix representations . The most important new concept we shall introduce is that of an invariant space . In the ...
... invariant methods ; that is , by making use only of the intrinsic properties of linear trans- formations and not of their matrix representations . The most important new concept we shall introduce is that of an invariant space . In the ...
Page 281
... invariant space of L. Hence , by Theorem 9.5.4 , L effects an automorphism of U and annihilates a certain complement u ' of U. Let { x1 , ... , x , } be a basis of U and { X + 1 , ... , X } a basis of ' . Then X = { X1 , ... , X ,, X + ...
... invariant space of L. Hence , by Theorem 9.5.4 , L effects an automorphism of U and annihilates a certain complement u ' of U. Let { x1 , ... , x , } be a basis of U and { X + 1 , ... , X } a basis of ' . Then X = { X1 , ... , X ,, X + ...
Table des matières
PART | 3 |
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 C₁ 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 isomorphic linear combination 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 nxn matrix obtain orthogonal matrix positive definite possesses proof of Theorem prove quadratic form quadric rank reduces 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 w₁ write x₁ y₁ zero α₁