An Introduction to Linear AlgebraClarendon Press, 1963 - 440 pages |
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Page 223
Leonid Mirsky. THEOREM 8.1.2 . A real matrix is orthogonal if and only if its columns ( or rows ) form an orthonormal ... orthogonal matrices VIII , § 8.1 223 ORTHOGONAL MATRICES.
Leonid Mirsky. THEOREM 8.1.2 . A real matrix is orthogonal if and only if its columns ( or rows ) form an orthonormal ... orthogonal matrices VIII , § 8.1 223 ORTHOGONAL MATRICES.
Page 233
... orthogonal substitutions preserve separation and may therefore be expected to occur in the analysis of rigid motion . In this section and the next we shall discuss in some detail the relation between orthogonal matrices and rotations ...
... orthogonal substitutions preserve separation and may therefore be expected to occur in the analysis of rigid motion . In this section and the next we shall discuss in some detail the relation between orthogonal matrices and rotations ...
Page 425
... matrix A be called -orthogonal if the substitution x = Ay leaves the quadratic form xTx invariant . Show that the most general form of an -orthogonal matrix is A = M - 1PM , where M is a certain fixed matrix and P an arbitrary ...
... matrix A be called -orthogonal if the substitution x = Ay leaves the quadratic form xTx invariant . Show that the most general form of an -orthogonal matrix is A = M - 1PM , where M is a certain fixed matrix and P an arbitrary ...
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