An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 229
... Unitary matrices In considering complex matrices , it is desirable to generalize the notion of orthogonality . DEFINITION 8.2.1 . The complex matrix U is UNITARY if UTU = I. ( 8.2.1 ) Thus a real unitary matrix is simply an orthogonal ...
... Unitary matrices In considering complex matrices , it is desirable to generalize the notion of orthogonality . DEFINITION 8.2.1 . The complex matrix U is UNITARY if UTU = I. ( 8.2.1 ) Thus a real unitary matrix is simply an orthogonal ...
Page 231
... 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 ...
Page 248
... unitary matrix , and evaluate its determinant . 2. Let α1 , ... , an be complex numbers of modulus 1. Show that , if the rows of a unitary matrix are multiplied by a1 , ... , a respectively , then the resulting matrix is again unitary ...
... unitary matrix , and evaluate its determinant . 2. Let α1 , ... , an be complex numbers of modulus 1. Show that , if the rows of a unitary matrix are multiplied by a1 , ... , a respectively , then the resulting matrix is again unitary ...
Table des matières
PART | 3 |
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 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