An Introduction to Linear AlgebraClarendon Press, 1972 - 440 pages |
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Page 84
... identities such as these are useful since they help to reduce the manipulation of matrices to a purely mechanical ... Identity ( ii ) is proved similarly . Again , using ( 3.3.6 ) and ( i ) , we have ( ABC ) ;; = ( AB ) i * C * i = Ai ...
... identities such as these are useful since they help to reduce the manipulation of matrices to a purely mechanical ... Identity ( ii ) is proved similarly . Again , using ( 3.3.6 ) and ( i ) , we have ( ABC ) ;; = ( AB ) i * C * i = Ai ...
Page 98
... identity between scalar poly- nomials remains valid for the corresponding matrix polynomials . Thus , for instance , if ƒ1 , ... , ƒ are polynomials , and { ƒ1 ( x ) ƒ2 ( x ) + ƒ3 ( x ) } ƒ4 ( x ) = ƒ5 ( x ) —ƒ 。( x ) , then , for ...
... identity between scalar poly- nomials remains valid for the corresponding matrix polynomials . Thus , for instance , if ƒ1 , ... , ƒ are polynomials , and { ƒ1 ( x ) ƒ2 ( x ) + ƒ3 ( x ) } ƒ4 ( x ) = ƒ5 ( x ) —ƒ 。( x ) , then , for ...
Page 99
... identity enables us to define rational functions of A. ( 3.7.7 ) DEFINITION 3.7.2 . Let f ( x ) , g ( x ) be scalar polynomials and let A be any square matrix such that g ( A ) | 0. Then the matrix appear- ing on either side of ( 3.7.7 ) ...
... identity enables us to define rational functions of A. ( 3.7.7 ) DEFINITION 3.7.2 . Let f ( x ) , g ( x ) be scalar polynomials and let A be any square matrix such that g ( A ) | 0. Then the matrix appear- ing on either side of ( 3.7.7 ) ...
Table des matières
DETERMINANTS VECTORS MATRICES | 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 C₁ canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix 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 values vector space view of Theorem write x₁ xTAx y₁ zero α₁