Elementary Theory of Matrices |
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Page 16
... fact that this one to one correspondence was set up by means of a particular coordinate system { x } and that as we pass from one coordinate system to another the same linear transformation may correspond to several matrices and one ...
... fact that this one to one correspondence was set up by means of a particular coordinate system { x } and that as we pass from one coordinate system to another the same linear transformation may correspond to several matrices and one ...
Page 42
... fact is also easy to derive from the general commutativity theorem of ( III.8.4 . ) we have AB = f ( A ) 2f ( B ) 2 ... fact that if C is a Hermitian transformation and pλ ) and qλ ) are any two polynomials then A = p ( C ) and E = q ( C ) ...
... fact is also easy to derive from the general commutativity theorem of ( III.8.4 . ) we have AB = f ( A ) 2f ( B ) 2 ... fact that if C is a Hermitian transformation and pλ ) and qλ ) are any two polynomials then A = p ( C ) and E = q ( C ) ...
Page 74
... fact that the metric topology defined by the notion of convergence is complete : i.e. if x is a sequence of vectors for which xn xml- O as n , m , then there is a 1 unique vector x such that xx as n → ∞ . 2. The notion of convergence ...
... fact that the metric topology defined by the notion of convergence is complete : i.e. if x is a sequence of vectors for which xn xml- O as n , m , then there is a 1 unique vector x such that xx as n → ∞ . 2. The notion of convergence ...
Contents
Chapter II | 15 |
Linear transformations in vector spaces | 23 |
Isomorphism of matrices and transformations | 27 |
1 other sections not shown
Common terms and phrases
₁₁₁ absolute value algebraic arbitrary linear transformation arbitrary vector Ax,x Ax,y bilinear bound chapter commutative complete orthonormal set complex numbers conjugate coordinate system correspondence cyclic subspaces define definition denote dimensional direct product direct sum dual space E₁ easy to verify eigenvalues equation equivalent Ex,y fact follows form Ax Hence Hermitian transformations idempotent implies inner product invariant factors inverse irreducible polynomials isomorphism linear combination linear functions linear trans linear transformation linearly independent M₁ Markoff matrix minimal polynomial N₁ necessary and sufficient negative transformation normal transformations o.n. set obtain orthogonal basis orthogonal projections preceding section proof properties prove rational canonical form real numbers relation representation respectively result scalar multiplication Schwartz inequality spectral form sufficient condition tion transformations of rank unitary space unitary transformation vector space whence words write x₁ zero Σ₁