MATHEMATICS

E. A. TROITSKAYA

APPLICATION OF THE GENERAL THEORY OF APPROXIMATE METHODS TO THE STUDY OF THE PROBLEM OF DETERMINING EIGENVALUES AND EIGENVECTORS

(Presented by Academician V. I. Smirnov, 5 XI 1956)

We consider two completely continuous operators: \(A\) in a linear normed space \(X\) and \(\bar A\) in a complete linear normed space \(\bar X\), connected in the following way. In the space \(X\) there exists a subspace \(\widetilde X\), isomorphic to \(\bar X\). The isomorphism is carried out by means of a linear operation \(\varphi_0\), which has an inverse \(\varphi_0^{-1}\) and admits an extension \(\varphi\) to the whole space \(X\).

The following conditions are satisfied:

I. For every \(\tilde x \in \widetilde X\),
\[ \|\varphi A\tilde x-\bar A\varphi \tilde x\|\leq \varepsilon \|\tilde x\|. \]

II. For every \(x\in X\) one can find \(\tilde x\in \widetilde X\) such that
\[ \|Ax-\tilde x\|\leq \varepsilon_1\|x\|. \]

In the work of L. V. Kantorovich \((^1)\) the operators \(K=A-\lambda I\) and \(\bar K=\bar A-\lambda I\) are considered, and conditions are given whose fulfillment guarantees the existence of the operator \(\bar K^{-1}\), if \(K^{-1}\) exists, and a set of estimates.

Let a simple eigenvalue \(\lambda_0\) and an eigenvector \(x_0\) of the operator \(A\) be known, as well as an eigenvector \(f_0\) of the adjoint operator \(A^*\). We shall assume that \(f_0(x_0)=1\); consequently, the pair \(\lambda_0,x_0\) is a solution of the system
\[ Ax-\lambda x=0, \]
\[ f_0(x)-1=0. \]

If we introduce the space \(U\), whose elements are pairs
\[ u=\binom{x}{\lambda}\quad (x\in X;\ \lambda\text{ is a complex number};\ \|u\|^2=\|x\|^2+|\lambda|^2), \]
then this system can be written in the form of a single nonlinear functional equation in the space \(U\):
\[ P\binom{x}{\lambda}=\binom{Ax-\lambda x}{f_0(x)-1}=0. \tag{1} \]

We normalize the eigenvector of the operator \(\bar A\) in the following way:
\[ f_0(\varphi_0^{-1}\bar x)=1. \]
Then to equation (1) one can put in correspondence the following equation in the space \(\bar U\) of elements
\[ \bar u=\binom{\bar x}{\lambda}\quad (\bar x\in \bar X;\ \lambda\text{ is a complex number}): \]
\[ \bar P\binom{\bar x}{\lambda}= \binom{\bar A\bar x-\lambda\bar x}{f_0(\varphi_0^{-1}\bar x)-1}=0. \tag{2} \]

The space \(U\) contains the subspace
\[ \widetilde U=\left\{\binom{\tilde x}{\lambda}\right\}\quad (\tilde x\in \widetilde X), \]
isomorphic to \(\bar U\). The isomorphism is carried out by means of the operation
\[ \Phi_0\binom{x}{\lambda}=\binom{\varphi x}{\lambda}, \]

having the inverse

\[ \Phi_0^{-1}\binom{\bar x}{\lambda}=\binom{\varphi_0^{-1}\bar x}{\lambda}; \qquad \Phi\binom{x}{\lambda}=\binom{\varphi x}{\lambda} \]

is an extension of \(\Phi_0\) to the whole space \(U\).

Thus, the determination of the eigenvalue \(\bar\lambda_0\) and eigenvector \(\bar x_0\) of the operator \(\bar A\) is reduced to solving a nonlinear equation. In solving it one may use the analogue, proposed by L. V. Kantorovich, of Newton’s method for solving functional equations [1], for which the most essential requirement is the existence of an operation inverse to the derivative operation \(\bar P'_{\bar u_0}\), computed for the initial approximation of Newton’s method. Taking the element

\[ \bar u_0=\binom{\varphi x_0}{\lambda_0}, \]

all the conditions for its applicability are fulfilled.

The derivatives of the operators \(\bar P_{\varphi x_0,\lambda_0}\) and \(P'_{x_0,\lambda_0}\) have the form:

\[ \bar P'_{\varphi x_0,\lambda_0}\binom{\bar x}{\lambda} = \binom{\bar A\bar x-\lambda_0\bar x-\lambda\varphi x_0} {f_0\left(\varphi_0^{-1}\bar x\right)}, \]

\[ P'_{x_0,\lambda_0}\binom{x}{\lambda} = \binom{Ax-\lambda_0x-\lambda x_0} {f_0(x)}, \]

and it is easy to show that the latter operator has the inverse

\[ \left(P'_{x_0,\lambda_0}\right)^{-1}\binom{y}{\mu} = \binom{Ry+\mu x_0}{-f_0(y)}, \qquad \left\|\left(P'_{x_0,\lambda_0}\right)^{-1}\right\|^2 \le \|R\|^2+\|x_0\|^2+\|f_0\|^2. \]

Here the following notation has been adopted: \(R\) is the resolvent operator \(R=(A-\lambda_0 I)^{-1}\), defined and bounded on the set of those \(x\) for which \(f_0(x)=0\), and equal to zero on the eigensubspace.

The operators \(\bar P_{\varphi x_0,\lambda_0}\) and \(P'_{x_0,\lambda_0}\) are connected by conditions analogous to conditions I and II:

I.

\[ \begin{aligned} &\left\| \Phi P'_{x_0,\lambda_0}\binom{\widetilde x}{\lambda} - \bar P'_{\varphi x_0,\lambda_0} \Phi\binom{\widetilde x}{\lambda} \right\| \\ &= \left\| \binom{\varphi A\widetilde x-\lambda_0\varphi\widetilde x-\lambda\varphi x_0} {f_0(\widetilde x)} - \binom{\bar A\varphi\widetilde x-\lambda_0\varphi\widetilde x-\lambda\varphi x_0} {f_0\left(\varphi_0^{-1}\varphi\widetilde x\right)} \right\| \\ &= \left\|\varphi A\widetilde x-\bar A\varphi\widetilde x\right\| \le \varepsilon\|\widetilde x\| \le \varepsilon\left\|\binom{\widetilde x}{\lambda}\right\|. \end{aligned} \]

II. It is necessary to approximate, by elements of the space \(\widetilde U\), the elements

\[ P'\binom{x}{\lambda}+\lambda_0\binom{x}{\lambda} = \binom{Ax-\lambda x_0}{f_0(x)+\lambda\lambda_0}. \]

Find \(\widetilde x_1\) so that \(\|Ax-\widetilde x_1\|\le\varepsilon_1\|x\|\), and \(\widetilde x_0\) so that \(\|Ax_0-\widetilde x_0\|\le\varepsilon_1\|x_0\|\). Then for

\[ \widetilde x=\widetilde x_1-\frac{\lambda}{\lambda_0}\widetilde x_0 \quad\text{and}\quad \widetilde\lambda=-f_0(x)-\lambda\lambda_0 \]

we have:

\[ \left\| \binom{Ax-\lambda x_0}{f_0(x)+\lambda\lambda_0} - \binom{\widetilde x}{\widetilde\lambda} \right\| = \left\|Ax-\lambda x_0-\widetilde x\right\| \le \left\|Ax-\widetilde x_1\right\| + \left\|\lambda x_0-\frac{\lambda}{\lambda_0}\widetilde x_0\right\| \le \]

\[ \le \varepsilon_1\|x\| + \left\| \frac{\lambda}{\lambda_0}Ax_0-\frac{\lambda}{\lambda_0}\widetilde x_0 \right\| \le \varepsilon_1 \left( 1+\frac{\|x_0\|^2}{|\lambda_0|^2} \right)^{1/2} \left\|\binom{x}{\lambda}\right\|. \]

If the quantity

\[ q= \left\{ \varepsilon+ \varepsilon_1 \sqrt{ 1+\frac{\|x_0\|^2}{|\lambda_0|^2} } \left(\varepsilon+\|\Phi\| \sqrt{\|A-\lambda_0 I\|^2+\|f_0\|^2+\|x_0\|^2} \right) \right\} \times \]

\[ \times \|\Phi_0^{-1}\| \sqrt{\|R\|^2+\|f_0\|^2+\|x_0\|^2} <1, \]

then, according to the theorems of the general theory of approximate methods \((^1)\), from the existence of \((P'_{x_0,\lambda_0})^{-1}\) there follows the existence of \((\bar P'_{\varphi x_0,\lambda_0})^{-1}\), and

\[ \left\|(\bar P'_{\varphi x_0,\lambda_0})^{-1}\right\| \le \frac{ \left(1+\sqrt{1+\frac{\|x_0\|^2}{|\lambda_0|^2}}\right) \|\Phi\|\,\|\Phi_0^{-1}\|\sqrt{\|R\|^2+\|f_0\|^2+\|x_0\|^2} }{1-q} \equiv B_0 . \]

Let us estimate \(\bar P\binom{\varphi x_0}{\lambda_0}\):

\[ \bar P\binom{\varphi x_0}{\lambda_0} = \binom{\bar A\varphi x_0-\lambda_0\varphi x_0}{f_0(\varphi_0^{-1}\varphi x_0)-1} = \binom{\bar A\varphi x_0-\varphi A x_0}{f_0(\varphi_0^{-1}\varphi x_0)-f_0(x_0)} . \]

One can find \(\tilde x_0\in\tilde X\) such that

\[ \left\|x_0-\frac{\tilde x_0}{\lambda_0}\right\| = \left\|\frac{Ax_0}{\lambda_0}-\frac{\tilde x_0}{\lambda_0}\right\| \le \varepsilon_1\left|\frac{\|x_0\|}{\lambda_0}\right|. \]

Denote \(\tilde x'_0=\dfrac{\tilde x_0}{\lambda_0}\); \(\|\tilde x'_0\|\le\left(1+\dfrac{\varepsilon_1}{|\lambda_0|}\right)\|x_0\|\);

\[ \begin{aligned} \|\bar A\varphi x_0-\varphi A x_0\| &= \|\bar A\varphi x_0-\bar A\varphi\tilde x'_0-\bar A\varphi\tilde x'_0+\varphi A\tilde x'_0-\varphi A\tilde x'_0-\varphi A x_0\| \\ &\le \varepsilon_1\left|\frac{\|x_0\|}{\lambda_0}\right|(\|\bar A\varphi\|+\|\varphi A\|) +\varepsilon\|\tilde x'_0\| \\ &\le \left[\varepsilon_1(\|\bar A\varphi\|+\|\varphi A\|+\varepsilon)+\varepsilon|\lambda_0|\right]\left|\frac{\|x_0\|}{\lambda_0}\right|. \end{aligned} \]

Further,

\[ |\,\varphi_0^{-1}\varphi x_0-x_0\,| \le \|\varphi_0^{-1}\varphi x_0-\varphi_0^{-1}\varphi\tilde x'_0\| +\|\tilde x'_0-x_0\| \le (\|\varphi_0^{-1}\varphi\|+1)\|\tilde x'_0-x_0\| \le \varepsilon_1(1+\|\varphi_0^{-1}\varphi\|)\left|\frac{\|x_0\|}{\lambda_0}\right|. \]

Hence

\[ \left\|\bar P\binom{\varphi x_0}{\lambda_0}\right\|^2 \le \left|\frac{\|x_0\|^2}{\lambda_0}\right| \left\{ \left[\varepsilon_1(\varepsilon+\|\bar A\varphi\|+\|\varphi A\|)+\varepsilon|\lambda_0|\right]^2 +\varepsilon_1^2(1+\|\varphi_0^{-1}\varphi\|)^2\|f_0\|^2 \right\} \equiv \eta_0^2 . \]

The second derivative \(P''_u\) is bounded everywhere,

\[ \|P''_u\|\le\sqrt{3/2}. \]

It remains necessary that the condition \(h_0=\sqrt{3/2}\,B_0^2\eta_0\le 1/2\) be fulfilled, and then application of the theorems of Newton’s method \((^1)\) gives the following result:

Theorem 1. Let \(\lambda_0\) be a simple eigenvalue of the operator \(A\); \(x_0\) its eigenvector; \(f_0\) an eigenvector of the adjoint operator, with \(f_0(x_0)=1\).

If conditions I and II are satisfied,

\[ q= \left\{ \varepsilon+\varepsilon_1\sqrt{1+\frac{\|x_0\|^2}{|\lambda_0|^2}} \left(\varepsilon+\|\Phi\|\sqrt{\|A-\lambda_0 I\|^2+\|f_0\|^2+\|x_0\|^2}\right) \right\} \cdot \|\Phi_0^{-1}\|\sqrt{\|R\|^2+\|f_0\|^2+\|x_0\|^2} <1; \]

\[ \begin{aligned} h_0 &= \sqrt{\frac32}\,\frac1{|\lambda_0|} \frac{ \left(1+\varepsilon_1\sqrt{1+\frac{\|x_0\|^2}{|\lambda_0|^2}}\right)^2 \|\Phi\|^2\|\Phi_0^{-1}\|^2 (\|R\|^2+\|f_0\|^2+\|x_0\|^2) }{(1-q)^2} \\ &\quad{}\times \left\{ \bigl[\varepsilon|\lambda_0|+\varepsilon_1(\varepsilon+\|\bar A\varphi\|+\|\varphi A\|)\bigr]^2 +\varepsilon_1^2(1+\|\varphi_1^{-1}\varphi\|)^2\|f_0\|^2 \right\}^{1/2} \le \frac12 , \end{aligned} \]

then in the domain

\[ \left\|\binom{\bar x}{\lambda}-\binom{\varphi x_0}{\lambda_0}\right\| \le \frac{1-\sqrt{1-2h_0}}{h_0}\,\eta_0 \]

the operator \(\bar A\) has a unique eigenvalue \(\bar\lambda_0\) and a corresponding unique eigenvector \(\bar x_0\) such that \(f_0(\varphi_0^{-1}x_0)=1\).

The closeness of the solutions can be estimated in the form

\[ \left\| \binom{\varphi_0^{-1}\bar x_0}{\bar\lambda_0} - \binom{x_0}{\lambda_0} \right\| \le \frac{1-\sqrt{1-2h_0}}{h_0}\,\eta_0\|\Phi_0^{-1}\| + \varepsilon_1(1+\|\Phi_0^{-1}\|\|\Phi\|)\left|\frac{\|x_0\|}{\lambda_0}\right|. \]

Quite analogously, the second result is obtained:

Theorem 2. Let \(\overline{\lambda}_0\) be a simple eigenvalue of the operator \(\overline{A}\); \(\overline{x}_0\) its eigen-element; \(f_0\) an eigen-element of the adjoint operator. If conditions I and II are satisfied,

\[ r=\frac{\varepsilon_1}{|\overline{\lambda}_0|} \left( 1+\|\Phi_0^{-1}\|\,\|\Phi P_{\varphi_0^{-1}x_0,\lambda_0}^{\,\prime}\| \sqrt{\|\overline{R}\|^2+\|\overline{f}_0\|^2+\|\overline{x}_0\|^2} +2\varepsilon\|\Phi_0^{-1}\| \sqrt{\|\overline{R}\|^2+\|\overline{f}_0\|^2+\|\overline{x}_0\|^2} \right)<1, \]

\[ h_0= \sqrt{\frac{3}{2}\, \frac{ \left[ \|\Phi_0^{-1}(\overline{P}_{\overline{x}_0,\lambda_0}^{\,1})^{-1}\Phi\| + \frac{2}{|\overline{\lambda}_0|} \left(1+\|\Phi_0^{-1}\|\, \|\Phi P_{\varphi_0^{-1}x_0,\lambda_0}^{\,\prime}\| \sqrt{\|\overline{R}\|^2+\|\overline{x}_0\|^2+\|\overline{f}_0\|^2}\right) \right]^2 }{ (1-r)^2 } } \times \left[ \varepsilon_1+(\varepsilon+\varepsilon_1\|\varphi\|)\|\varphi_0^{-1}\|\,\|\overline{x}_0\| \right]\le \frac12, \]

then the operator \(A\) has, in the domain

\[ \left\| \binom{x}{\lambda} - \binom{\varphi_0^{-1}\overline{x}_0}{\overline{\lambda}_0} \right\| \le \frac{1-\sqrt{1-2h_0}}{h_0} \left[ \varepsilon_1+(\varepsilon+\varepsilon_1\|\varphi\|)\|\varphi_0^{-1}\|\,\|\varphi_0^{-1}\|\,\|\overline{x}_0\| \right] \]

a unique eigen-pair \((x_0,\lambda_0)\).

Leningrad Branch of the Mathematical Institute
named after V. A. Steklov
Academy of Sciences of the USSR

Received
1 XI 1956

REFERENCES CITED

1. L. V. Kantorovich, Uspekhi Mat. Nauk, 3, no. 6 (28) (1948).