An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Leonid Mirsky. THEOREM 8.2.1 . If U is unitary , then | det U | = 1 . THEOREM 8.2.2 . If U , V are unitary , then so are U , UT , U - 1 and UV . These results follow immediately from ( 8.2.1 ) . The next theorem involves the notion of ...
Leonid Mirsky. THEOREM 8.2.1 . If U is unitary , then | det U | = 1 . THEOREM 8.2.2 . If U , V are unitary , then so are U , UT , U - 1 and UV . These results follow immediately from ( 8.2.1 ) . The next theorem involves the notion of ...
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... unitary , then ( 8.2.3 ) is clearly satisfied for all x . Suppose , on the other hand , that ( 8.2.3 ) is satisfied ... unitary . COROLLARY 1. A ( complex ) matrix U is unitary if and only if | Ux − Uy | = | x − y | for all complex ...
... unitary , then ( 8.2.3 ) is clearly satisfied for all x . Suppose , on the other hand , that ( 8.2.3 ) is satisfied ... unitary . COROLLARY 1. A ( complex ) matrix U is unitary if and only if | Ux − Uy | = | x − y | for all complex ...
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... unitary matrices , hermitian matrices , and skew - hermitian matrices . EXERCISE 10.3.4 . Verify that the property of normality is invariant under unitary similarity transformations , i.e. if A is normal and U is unitary , then U - 1AU ...
... unitary matrices , hermitian matrices , and skew - hermitian matrices . EXERCISE 10.3.4 . Verify that the property of normality is invariant under unitary similarity transformations , i.e. if A is normal and U is unitary , then U - 1AU ...
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