Sentence examples for basis of eigenvectors of from inspiring English sources

Suppose ({varphi _j}_{j=0}^{n-1}) are an orthonormal basis of eigenvectors of A with (Avarphi _j = E_jvarphi _j).

Let (V_{k}=operatorname{span}{v_{1},ldots,v_{k},v_{k}}^{bot}) and ({v_{j}}_{jgeq1}) be an orthonormal basis of eigenvectors of the operator (mathfrak{L}).

Let ({ |{phi}_{1} rangle,|{phi}_{2} rangle,ldots,|{phi}_{n} rangle }) be a basis of eigenvectors of ρ, corresponding to the eigenvalues ({ lambda_{1},ldotsa_{2},lambda_{nmbda_{n} }).

end{aligned} Then begin{aligned} Az=sum_{n=1} ^{infty}n^{2} z,z_{n}) z_{n}, end{aligned} where (z_{n}(t)=sqrt{frac{2}{pi}}sin(nt)), (n=1,2,ldots) , constitute the orthogonal basis of eigenvectors of A. It is well known that A generates a compact, analytic semigroup ({T t): tgeq0}) in H (see Pazy [34]).

Then (Aomega=sum_{n=1}^{infty}-n^{2}langleomega,e_{n}rangle e_{n}), (omegain D(A)), where (e_{n} z)=sqrt{frac{2}{pi}}sin{nz}) is the orthogonal basis of eigenvectors of A. Clearly, A generates a compact analytic semigroup ({S t)}_{t>0}) in H and it can be written by (S t)omega=sum_{n=1}^{infty}e^{-n^{2}t}langle omega,e_{n}rangle e_{n}), (omegain X).

Furthermore, the authors use linear combinations of and matrices as to furnish the basis of eigenvectors for the DFrFT matrix.

Then there exists a Hilbert basis composed of eigenvectors of (mathcal{T}).

A Cartan subalgebra of a Lie algebra (mathfrak {g}_0) is a maximal commutative subalgebra (mathfrak {h} subset mathfrak {g}_0) such that (mathfrak {g}_0) (or its complexification if (mathfrak {g}_0) is a real Lie algebra) has a basis consisting of eigenvectors of (mathrm {ad}(H)) for all (H in mathfrak {h},).

We assume that is a bounded self-adjoint operator with distinct eigenvalues,, and is an orthonormal basis of consisting of eigenvectors of corresponding to eigenvalues, respectively.

Therefore, applying Lemma 5.1, we know that (L^{2}([0,mathbbhbb {R})) admits a Hilbert basis ({phi_{n}}) consisting of eigenvectors of (mathcal{T}) with corresponding eigenvalues (mu_{n}).

Note that there exists a complete orthonormal basis ({e_{n}}_{nin N}) of eigenvectors of A with (e_{n} z)=sqrt{frac{2}{pi}}sin nz)), (n=1,2,ldots) , and A generates a strongly continuous semigroup ({S t),tgeq0}) which is compact, analytic and self-adjoint [6, 7].