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1
A Primer of Abstract Algebra
Let T be a linear operator on the n-dimensional inner product space V over C.
Then T is unitarily diagonalizable if and only ifT is normal. Proof. If T is unitarily
diagonalizable, then with respect to some orthonormal basis, T is represented by
a ...
2
Further Pure Mathematics
1 2 2 1 Eigenvalues X = 3, — 1. Orthogonally diagonalizable. In examples (a), (b)
and (g), the eigenvalues are real and distinct. In each case the matrix is
diagonalizable. This is true in general and it is our first general result: Result 1 A
square ...
Linda Bostock, Suzanne Chandler, Colin Patrick Rourke, 1982
3
Elementary Linear Algebra
We are now ready to show that symmetric operators and orthogonally
diagonalizable operators are really the same. Theorem 6.20 Let V be a nontrivial
subspace of Rn, and letL be a linear operator on V. Then L is orthogonally
diagonalizable ...
Stephen Andrilli, David Hecker, 2010
4
Observation and Control for Operator Semigroups
In this section we introduce diagonalizable operators, which can be described
entirely in terms of their eigenvalues and eigenvectors, thus having a very simple
structure. If a semigroup generator is diagonalizable, then so is the semigroup.
Marius Tucsnak, George Weiss, 2009
5
Elementary Linear Algebra
n n B A (b) The fact that an matrix has distinct eigenvalues does not guarantee
that is diagonalizable. A n A n n 38. (a) If is a diagonalizable matrix, then it has
linearly independent eigenvectors. n A (b) If an matrix is diagonalizable, then it
must ...
6
Indefinite Linear Algebra and Applications
General Perturbations and Stably Diagonalizable H-Unitary Matrices 199
satisfies the inequality vΩ(V) ≥ ∑ λ∈Ω |sig ( PRλ(U)HPRλ(U) ) |, (9.8.15) and in
every (open) neighborhood U⊂U U of U there exists an H-unitary V for which the
...
Israel Gohberg, Peter Lancaster, Leiba Rodman, 2006
7
Lie Groups and Invariant Theory
Clearly, two minimal 5-containing regular subgroups Ri and R( are conjugate in
G if and only if the corresponding diagonalizable subgroups D, and D[ are
conjugate in Z. Thus, Di is defined up to conjugation by an element of Z. We can
say ...
Ėrnest Borisovich Vinberg, 2005
8
Engineering Mathematics: Vol II; B.Sc. (Engg.), B.E., ...
7-2-36 Diagonalizable Operators. Definition. Let T:V—>V be a linear operator on
the finite dimensional vector space V then T is said to be diagonalizable if there is
a basis for V each vector of which is a characteristic vector of T. Matrix of a ...
9
Handbook of Linear Algebra
A is diagonalizable if and only if |a(A)| = n and A is nondefective. 23. If A has n
distinct eigenvalues, then A is diagonalizable. 24. A is diagonalizable if and only
if qA(x) can be factored into distinct linear factors. 25. If A is diagonalizable, then
so ...
10
Gradings on Simple Lie Algebras
On the other hand, Q = ηΓ(GD), so Γ is a fine grading with universal group G. If Γ
were a φ-grading for some φ, then we could construct an extension ̃G of G by Z2
as in Lemma 3.22 and hence a diagonalizable subgroup- scheme ̃Q of Aut(R) ...
Alberto Elduque, Mikhail Kochetov, 2013
