MIROSLAV FIEDLER, VLASTIMIL PTÁK
(MIROSLAV FIEDLER, VLASTIMIL PTÁK)

ESTIMATES AND ITERATIVE METHODS FOR FINDING A SIMPLE EIGENVALUE OF AN ALMOST DECOMPOSABLE MATRIX

(Presented by Academician S. L. Sobolev on 6 III 1963)

Let \(X\) be a complex linear space of dimension \(n\). We shall regard the vectors of this space as row vectors
\(x=(x_1,x_2,\ldots,x_n)\). A column vector with the same coordinates will be denoted by \(x'\). Then the vectors
\(y=(x_2,\ldots,x_n)\) range over a space \(Y\) of dimension \(n-1\). In \(Y\) we prescribe a norm \(g\) (for example,
\(g(y)=\sum_{i=2}^{n}|x_i|\)), to which there corresponds the norm of a matrix (operator) \(B\) in \(Y\),
\(g(B)=\sup g(yB)\) for \(g(y)=1\), and the “lower matrix norm” \(\hat g(B)=\inf g(yB)\) for \(g(y)=1\).
As usual, for column vectors we introduce the conjugate norm
\(g'(y')=\sup |yy'|\) for \(g(y)=1\).

For brevity we omit, in expressions of the type \(A-tE\), where \(t\) is a number and \(E\) is the identity matrix, the symbol \(E\). In studying the location of eigenvalues in the complex plane we use the following notation: if \(z_0\) is a complex number and \(\rho>0\), then by \(K(z_0,\rho)\) we denote the set of all complex numbers \(z\) for which \(|z-z_0|\leq \rho\).

The following problem is considered in the paper. Let

\[ A=\begin{pmatrix} a_{11} & a_1\\ a_2' & A_{22} \end{pmatrix} \]

be a block matrix (with blocks of dimensions \(1\) and \(n-1\)) such that at least one of the vectors \(a_1,a_2'\) is, in a certain sense, small. Then one may expect that some eigenvalue of the matrix \(A\) is close to \(a_{11}\). Precise results are given in the following theorem:

Theorem. Let

\[ A=\begin{pmatrix} a_{11} & a_1\\ a_2 & A_{22} \end{pmatrix} \]

be a block complex matrix (with blocks of dimensions \(1\) and \(n-1\)), and let \(B=A_{22}-a_{11}\) be nonsingular. Suppose that for numbers \(a_1,a_2,a_3\) satisfying the inequalities

\[ g(a_1)g'(a_2')/\hat g(B)\leq a_2,\qquad |a_1B^{-1}a_2'|\leq a_1,\qquad \hat g(B)\geq a_3, \]

the inequality

\[ \sqrt{a_1}+\sqrt{a_2}<\sqrt{a_3}. \]

is fulfilled.

Then (in the notation indicated in item 3):

1) In the circle \(U^*=K(a_{11};s_1+\omega)\) there lies exactly one eigenvalue of the matrix \(A\), which we shall denote by \(x\).

2) The following three iterative processes are meaningful and convergent:

\[ x_0=a_{11},\qquad x_{k+1}=a_{11}-a_1(A_{22}-x_k)^{-1}a_2',\qquad \lim x_k=x; \tag{1} \]

\[ y_0=a_{11},\qquad y_{k+1}=a_{11}-\frac{a_1B^{-1}a'_2}{1+a_1(A_{22}-y_k)^{-1}B^{-1}a'_2},\qquad \lim y_k=x; \tag{2} \]

\[ B_0=B,\qquad c'_0=B_0^{-1}a'_2,\qquad \omega_k=a_kc'_k,\qquad B_{k+1}=B_k+c_ka'_1+\omega_k; \tag{3} \]

\[ c'_{k+1}=-\omega_k B_{k+1}^{-1}c'_k; \]

putting

\[ z_k=a_{11}-\sum_{j=0}^{k-1}\omega_j \]

for \(k=1,2,\ldots\), we obtain that \(\lim z_k=x\).

In the notation indicated in item 3), the inclusions

\[ U^*\supset K(x_0;\rho_0)\supset K(x_1;\rho_1)\supset\cdots, \]

\[ U^*\supset K(y_0;r_0)\supset K(y_1;r_1)\supset\cdots, \]

\[ U^*\supset K(z_1;R_1)\supset K(z_2;R_2)\supset\cdots \]

are valid, and the point \(x\) is the unique point of intersection of each of these three chains.

3) List of notation:

\[ s_1=\alpha_3-\alpha_1+\alpha_2,\qquad s_2=\alpha_3+\alpha_1-\alpha_2,\qquad s_3=\alpha_3-\alpha_1-\alpha_2, \]

\[ w=\sqrt{s_1^2-4\alpha_2\alpha_3} =\sqrt{s_2^2-4\alpha_1\alpha_3} =\sqrt{s_3^2-4\alpha_1\alpha_2}>0, \]

\[ \lambda_i=\frac{s_i-w}{s_i+w}\quad \text{for } i=1,2,3,\qquad q=\frac{4\alpha_1\alpha_2}{s_3^2}<1, \]

\[ \rho_k=\frac{w\lambda_2\lambda_1^k}{1-\lambda_2\lambda_1^k},\qquad r_k=\frac{w\lambda_2\lambda_3^k}{1-\lambda_2\lambda_3^k},\qquad R_k=s_3\frac{1}{2^k}q^{2k-1}. \]

Corollary. Suppose

\[ A=\begin{pmatrix} a_{11} & a_1\\ a'_2 & A_{22} \end{pmatrix} \]

is a block complex matrix (with blocks of sizes \(1\) and \(n-1\)) and \(B=A_{22}-a_{11}\) is nonsingular. Let \(\gamma,\beta_2,\beta_2\) be real numbers such that

\[ \hat g(B)\geq \gamma>0,\qquad g(A_1)\leq \beta_1,\qquad g'(a'_2)\leq \beta_2. \]

Assume that

\[ \gamma^2>4\beta_1\beta_2. \]

Then the circle

\[ K\left(a_{11};\frac12\left(\gamma-\sqrt{\gamma^2-4\beta_1\beta_2}\right)\right) \]

contains exactly one eigenvalue of the matrix \(A\).

This corollary is a strengthening of theorem (5,5) from paper (1).

The proof of the theorem is based on the following simple lemma:

Lemma. Suppose

\[ A=\begin{pmatrix} a_{11} & a_1\\ a'_2 & A_{22} \end{pmatrix} \]

is a square block complex matrix with blocks of sizes \(1\) and \(n\), and suppose \(\lambda\) does not belong to the spectrum of the matrix \(A_{22}\).

Then

\[ \det(A-\lambda)=\left(a_{11}-\lambda-a_1(A_{22}-\lambda)^{-1}a'_2\right)\det(A_{22}-\lambda), \]

so that \(\lambda\) is an eigenvalue of the matrix \(A\) if and only if

\[ a_{11}-\lambda=a_1(A_{22}-\lambda)^{-1}a'_2. \]

Using this lemma, for \(|t|<\hat g(B)\) we introduce the function

\[ f(t)=-a_1(B-t)^{-1}a'_2. \]

It follows from the lemma that (under the condition that the matrix \(A_{22}-\lambda\) is nonsingular) the number \(\lambda\) is an eigenvalue of the matrix \(A\) if and only if \(\lambda-a_{11}=f(\lambda-a_{11})\). Process (1) can evidently be rewritten in the form

\[ x_0-a_{11}=0,\quad x_{k+1}-a_{11}=f(x_k-a_{11}). \]

Analyzing this process, we see that it converges to an eigenvalue of the matrix \(A\) lying in any \(K(x_k;\rho_k)\). Since \(f(t_1)-f(t_2)=(t_1-t_2)a_1(B-t_1)^{-1}(B-t_2)^{-1}a_2\), in \(U^*\) there lies no more than one eigenvalue of the matrix \(A\).

The proof of convergence and of the estimates for processes (2) and (3) is carried out analogously.

Mathematical Institute
Czechoslovak Academy of Sciences
Prague, Czechoslovak Socialist Republic

Received
28 II 1963

CITED LITERATURE

1. M. Fiedler, V. Ptak, Czechoslovak Mathematical Journal, 12 (87), 558 (1962).