An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 161
... Theorem 5.6.2 gives an upper bound for the rank of the product of two matrices . To obtain a lower bound we first need a preliminary result . = THEOREM 5.6.4 . The vectors of the form Bx , subject to the condition ABX 0 , constitute a ...
... Theorem 5.6.2 gives an upper bound for the rank of the product of two matrices . To obtain a lower bound we first need a preliminary result . = THEOREM 5.6.4 . The vectors of the form Bx , subject to the condition ABX 0 , constitute a ...
Page 292
... Theorem 10.1.1 naturally leads us to inquire in what order the characteristic roots of A appear on the diagonal of a canonical form . The next theorem shows that for diagonal canonical forms this order may be preassigned arbitrarily .
... Theorem 10.1.1 naturally leads us to inquire in what order the characteristic roots of A appear on the diagonal of a canonical form . The next theorem shows that for diagonal canonical forms this order may be preassigned arbitrarily .
Page 411
... theorem is Theorem 13.4.3 is mainly valuable as an existence theorem , for the actual construction of the reducing matrix U as described in the proof above is extremely tedious in all but the simplest numerical cases . The reason for ...
... theorem is Theorem 13.4.3 is mainly valuable as an existence theorem , for the actual construction of the reducing matrix U as described in the proof above is extremely tedious in all but the simplest numerical cases . The reason for ...
Table des matières
PART | 1 |
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 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