An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 76
... scalar matrices are the unit matrix and the square zero matrix . We shall presently see that scalar matrices behave essentially like scalars . DEFINITION 3.2.6 . An UPPER ( LOWER ) TRIANGULAR MATRIX is a square matrix all of whose ...
... scalar matrices are the unit matrix and the square zero matrix . We shall presently see that scalar matrices behave essentially like scalars . DEFINITION 3.2.6 . An UPPER ( LOWER ) TRIANGULAR MATRIX is a square matrix all of whose ...
Page 92
... scalar matrices and that of scalars . We have , in fact , ( x + 8 ) I , aI ± ẞI = aI.ẞI = ( αẞ ) I , ( αI ) -1 = α - 1I ( α 0 ) . Thus there is a complete correspondence between the scalars and the scalar matrices , each scalar a being ...
... scalar matrices and that of scalars . We have , in fact , ( x + 8 ) I , aI ± ẞI = aI.ẞI = ( αẞ ) I , ( αI ) -1 = α - 1I ( α 0 ) . Thus there is a complete correspondence between the scalars and the scalar matrices , each scalar a being ...
Page 98
... scalar variable is known as a scalar polynomial . We recall that two scalar polynomials f ( x ) , g ( x ) are said to be identically equal if f ( x ) = g ( x ) for all values of x . THEOREM 3.7.1 . If the polynomials f ( x ) , g ( x ) ...
... scalar variable is known as a scalar polynomial . We recall that two scalar polynomials f ( x ) , g ( x ) are said to be identically equal if f ( x ) = g ( x ) for all values of x . THEOREM 3.7.1 . If the polynomials f ( x ) , g ( x ) ...
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