An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 75
... square matrices from the discussion of determinants . When we wish to emphasize that we are not necessarily dealing with square matrices we speak of rectangular matrices . DEFINITION 3.2.3 . The ZERO MATRIX ( of type m × n ) is the mx n ...
... square matrices from the discussion of determinants . When we wish to emphasize that we are not necessarily dealing with square matrices we speak of rectangular matrices . DEFINITION 3.2.3 . The ZERO MATRIX ( of type m × n ) is the mx n ...
Page 96
... square matrices . Show that either of the relations AX AY , XB = YB implies X = Y. = The problem of determining which matrices are divisors of zero now naturally presents itself . We shall solve it here for the case of square matrices ...
... square matrices . Show that either of the relations AX AY , XB = YB implies X = Y. = The problem of determining which matrices are divisors of zero now naturally presents itself . We shall solve it here for the case of square matrices ...
Page 97
... square matrix , then ( AT ) -1 = ( A - 1 ) T . In other words , the operations of transposition and inversion can be carried out in either order . i.e. To prove the theorem we put B = A - 1 in ( 3.6.12 ) and obtain ( AA - 1 ) T = ( A ...
... square matrix , then ( AT ) -1 = ( A - 1 ) T . In other words , the operations of transposition and inversion can be carried out in either order . i.e. To prove the theorem we put B = A - 1 in ( 3.6.12 ) and obtain ( AA - 1 ) T = ( A ...
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