An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 93
... singular matrices , then so is AB ; and ( AB ) -1 B - 1A - 1 . = If A is non - singular , then A - 1 exists and A - 1A I = ( 3.6.6 ) ( 3.6.7 ) Hence the equation A - 1X = I is soluble and has X = A as a solution . In view of Theorem 3.6 ...
... singular matrices , then so is AB ; and ( AB ) -1 B - 1A - 1 . = If A is non - singular , then A - 1 exists and A - 1A I = ( 3.6.6 ) ( 3.6.7 ) Hence the equation A - 1X = I is soluble and has X = A as a solution . In view of Theorem 3.6 ...
Page 193
... singular ; ( ii ) a system of m homo- geneous linear equations in n > m unknowns always possesses a non - trivial solution . 11. Show that every non - singular 2 × 2 matrix can be represented as a product of matrices of the following ...
... singular ; ( ii ) a system of m homo- geneous linear equations in n > m unknowns always possesses a non - trivial solution . 11. Show that every non - singular 2 × 2 matrix can be represented as a product of matrices of the following ...
Page 376
... singular linear transformations . All these quadratic forms will represent , when equated to 0 , the same conic C , referred in each case to a suitable system of projective coordinates . It is principally for this reason that we are ...
... singular linear transformations . All these quadratic forms will represent , when equated to 0 , the same conic C , referred in each case to a suitable system of projective coordinates . It is principally for this reason that we are ...
Table des matières
PART | 3 |
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 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