An Introduction to Linear AlgebraClarendon Press, 1955 - 433 pages |
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Page 76
... DIAGONAL MATRIX is a square matrix all of whose elements outside the diagonal are equal to zero . ( ii ) A SCALAR MATRIX is a diagonal matrix all of whose diagonal elements are equal to one another . Thus a diagonal matrix of order n ...
... DIAGONAL MATRIX is a square matrix all of whose elements outside the diagonal are equal to zero . ( ii ) A SCALAR MATRIX is a diagonal matrix all of whose diagonal elements are equal to one another . Thus a diagonal matrix of order n ...
Page 318
Leonid Mirsky. if and only if X is similar to a diagonal matrix . Now clearly k fw ( A ) = Σ fw ( A¡ ) , and so 1⁄2 fa ( A1 ) = - 0 ω . for all w . Hence , by ( 10.6.1 ) , fw ( A ; ) = 0 for i = 1 , ... , k and all Each matrix A ; is ...
Leonid Mirsky. if and only if X is similar to a diagonal matrix . Now clearly k fw ( A ) = Σ fw ( A¡ ) , and so 1⁄2 fa ( A1 ) = - 0 ω . for all w . Hence , by ( 10.6.1 ) , fw ( A ; ) = 0 for i = 1 , ... , k and all Each matrix A ; is ...
Page 324
... diagonal matrix . = 24. Show that a matrix which is similar to a diagonal matrix possesses a critical principal minor . Deduce that the rank of a skew - hermitian matrix is even . 25. Let A be a unitary matrix , B a normal matrix , and ...
... diagonal matrix . = 24. Show that a matrix which is similar to a diagonal matrix possesses a critical principal minor . Deduce that the rank of a skew - hermitian matrix is even . 25. Let A be a unitary matrix , B a normal matrix , and ...
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