GENERALIZED SPECTRAL PROBLEM FOR TWO ARBITRARY MATRICES \(a_{(nn)}\), \(\varphi_{(nn)}\) AND ITS SOLUTION

B. F. KRUTOI

(Presented by the scientific seminar of the Department of Engineering and Computational Mathematics)

1. Suppose that we are given two arbitrary matrices \(a_{(nn)}\), \(\varphi_{(nn)}\), real or complex. Then the following generalized spectral problem may be posed:

Determine the sets (spectrum) of eigenvalues \(\lambda\) and eigenvectors \(g_{(n)}=\bar g\) for the pair of matrices \(a_{(nn)}\) and \(\varphi_{(nn)}\), satisfying the equation

\[ a_{(nn)}g_{(n)}=\lambda\varphi_{(nn)}g_{(n)}, \tag{1} \]

or, in operator form,

\[ A\bar g=\lambda\Phi\bar g, \tag{1*} \]

where \(A\) and \(\Phi\) are linear operators corresponding to the matrices \(a_{(nn)}\) and \(\varphi_{(nn)}\) in some reference basis, and \(\lambda\) and \(\bar g\) are the eigenvalues and eigenvectors of the operator pair \(A\) and \(\Phi\).

Problems of this kind arise in solving, in a closed domain \(\overline D^{(n)}=D^{(n)}+\Gamma\), linear operator equations

\[ L(x)u(x)=f(x),\qquad x=(x_1,x_2,\ldots,x_n), \tag{2} \]

with the boundary condition

\[ \mathcal L(x)u(x)\big|_{\Gamma}=h(x) \tag{3} \]

on the boundary \(\Gamma\) of the domain \(\overline D\), when it is required to find the eigenvalues \(\lambda\) and eigenfunctions \(w(x)\) of the operators \(L\) and \(\mathcal L\) satisfying the conditions:

\[ 1)\quad L(x)w(x)=\Theta,\qquad 2)\quad \mathcal L(x)w(x)\big|_{\Gamma}=h(x), \tag{4} \]

where \(\Theta\) is the zero operator. Then the solution of problem (4) by a direct method, in which the sought eigenfunction \(w(x)\) of the operators \(L\), \(\mathcal L\) is specified in the form of the expansion

\[ w(x)\approx\varphi(x)=\sum_{\nu=0}^{m} c_\nu\varphi_\nu(x). \tag{5} \]

with respect to \(m\) mutually independent reference functions \(\varphi_\nu(x)\), leads to the need first to solve the generalized spectral problem (1). Such a case, for example, was considered in my paper [1] in solving problem (4) by the least-squares method.

1. Let us proceed to the solution of the generalized spectral problem defined by the homogeneous equation (1) with unknowns \(\lambda\) and \(g_{(n)}\). Writing this homogeneous equation in the form

\[ (a_{(nn)}-\lambda \varphi_{(nn)})g_{(n)}=0_{(n)}, \tag{1a} \]

we first of all note that we satisfy (1a) by taking \(g=0\). If, however, we set as our goal to determine from (1a) nonzero values of the eigenvector \(g_{(n)}\), then we shall achieve this by equating to 0 the coefficient \((a_{(nn)}-\lambda \varphi_{(nn)})\) in equation (1a), which is equivalent to the condition

\[ |a_{(nn)}-\lambda \varphi_{(nn)}|=0 \tag{6} \]

for the determinant \(|a_{(nn)}-\lambda \varphi_{(nn)}|\) of the matrix coefficient \((a_{(nn)}-\lambda \varphi_{(nn)})\).

Expanding determinant (6), we obtain an algebraic equation of the \(n\)-th degree with unknown \(\lambda\)

\[ |a_{(nn)}-\lambda \varphi_{(nn)}|=\sum_{x=0}^{n} b_x \lambda^x=0. \tag{6*} \]

Solving this equation, we obtain \(n\) values \(\lambda_s\) of the unknown \(\lambda\), among which complex pairs of the form \(\lambda_\nu=\alpha_\nu+i\beta_\nu\), \(\lambda_{\nu+1}=\alpha_\nu-i\beta_\nu\) may also occur. Multiple roots \(\lambda_\beta\) are also possible among the \(n\) roots \(\lambda_s\) of the unknown \(\lambda\). Therefore the number \(r\) of distinct roots \(\lambda_\beta\) may also be less than \(n\), so that \(r \le n\).

Substituting the found \(r \le n\) distinct roots \(\lambda_\beta\) of equation (6*) into the system of equations (1a), we obtain a bundle of \(r\) systems of equations with unknown vectors \(\bar g_\beta = g_{(n)\beta}\), the eigenvectors of the operator pair \(A,\Phi\):

\[ a_{(nn)}g_{(n)\beta}=\varphi_{(nn)}g_{(n)\beta}\lambda_\beta, \qquad (\beta=1,2,\ldots,r) \tag{7} \]

which can also be represented in matrix form as

\[ a_{(nn)}g_{(nr)}=\varphi_{(nn)}g_{(nr)}\lambda_{(rr)}, \tag{7a} \]

where \(\lambda_{(nn)}\) is a diagonal matrix

\[ \lambda_{(rr)}= \begin{bmatrix} \lambda_{11} & 0 & \cdots & 0 & \cdots & 0\\ 0 & \lambda_{22} & \cdots & 0 & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & \lambda_{33} & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ 0 & 0 & \cdots & 0 & \cdots & \lambda_{rr} \end{bmatrix}, \qquad \lambda_{\beta\beta}=\lambda_\beta. \tag{8} \]

As for the operator equation (1*), it now takes the following form:

\[ A\bar g_\beta=\lambda_\beta \Phi \bar g_\beta, \qquad (\beta=1,2,\ldots,r\le n). \tag{7*} \]

The obtained homogeneous matrix equation (7*), where, in addition to the given \(a_{(nn)}\) and \(\varphi_{(nn)}\), the diagonal matrix is now also known ...

\(\lambda_{(rr)}\), gives, for the sought matrix \(g_{(nr)}\), a value that is not completely determined. This is explained by the fact that each matrix-vector equation (7), with given \(\beta\) and sought vector \(g_{(n)\beta}\), entering relation (7a), is homogeneous. Therefore one of the components \(g_{\alpha\beta}\) in the \(n\)-dimensional vector \(g_{(n)\beta}\) remains undetermined, and we shall take

\[ g_{(n)\beta}=1,\quad (\beta=1,\,2,\,\ldots,\,r\le n). \tag{9} \]

The procedure considered above for solving the generalized spectral problem (1) convinces us that, for the eigenvalues \(\lambda\) of the operator pair \(A,\Phi\), we always obtain exactly \(n\) values \(\lambda_s\), since we find them from an algebraic equation of degree \(n\) \((6^*)\). True, a part \(r\le n\) of these roots \(\lambda\), denoted by \(\lambda_\beta\), may have multiplicity \(\partial_\beta\), where \(1\le \partial_\beta\le n\). Thus,

\[ \{\lambda_1,\lambda_2,\ldots,\lambda_\beta,\ldots,\lambda_r\}\le \{\lambda_1,\lambda_2,\ldots,\lambda_s,\ldots,\lambda_n\}. \tag{10} \]

As for the set of eigenvectors \(\bar g=\bar g_{(n)}\) for the operator pair \(A,\Phi\), their determination from relation (7a) inevitably gives only \(r\le n\) distinct values \(\bar g_\beta=g_{(n)\beta}\) of these vectors, corresponding to \(n\le r\) distinct values \(\lambda_\beta\) of the quantity \(\lambda\).

1. We next pose the problem: for any value \(r\le n\) of the number of eigenvectors \(\bar g_\beta=g_{(n)\beta}\) for the operator pair \(A,\Phi\), to determine another \(n-r\) associated vectors \(f_j\), \((j=r+1,r+2,\ldots,n)\), which would be linearly independent among themselves and with the \(r\) vectors \(\bar g_\beta\). Such a problem cannot be solved uniquely, but, generalizing somewhat Jordan’s proposal [1], we shall find one of its simplest solutions.

For this purpose, with each eigenvector \(\bar g_\beta\), generated by the eigenvalue \(\lambda_\beta\) of multiplicity \(\partial_\beta\), we associate another \(\partial_\beta-1\) associated vectors \(\bar g_{\beta\alpha}\), where \(\bar g_{\beta1}=\bar g_\beta\) and \(\alpha=2,3,\ldots,\partial_\beta\). We relate this set of \(\partial_\beta\) vectors \(\bar g_\beta=\bar g_{\beta1}, \bar g_{\beta2}, \bar g_{\beta3},\ldots,\bar g_{\beta\partial_\beta}\) by a system of \(\partial_\beta\) vector linearly independent equations of the following simplest structure:

\[ \left\{ \begin{aligned} 1)\;& \bar A\bar g_{\beta1} =\lambda_\beta\Phi\bar g_{\beta1} +0\,\bar g_{\beta2} +0\,\bar g_{\beta3} +\ldots +0\,\bar g_{\beta,\partial_\beta-1} +0\,\bar g_{\beta,\partial_\beta},\\ 2)\;& \bar A\bar g_{\beta2} =1\,\bar g_{\beta1} +\lambda_\beta\Phi\bar g_{\beta2} +0\,\bar g_{\beta3} +\ldots +0\,\bar g_{\beta,\partial_\beta-1} +0\,\bar g_{\beta,\partial_\beta},\\ 3)\;& \bar A\bar g_{\beta3} =0\,\bar g_{\beta1} +1\,\bar g_{\beta1} +\lambda_\beta\Phi\bar g_{\beta3} +\ldots +0\,\bar g_{\beta,\partial_\beta-1} +0\,\bar g_{\beta,\partial_\beta},\\ &\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ \partial_\beta)\;& A\bar g_{\beta,\partial_\beta} =0\,\bar g_{\beta1} +0\,\bar g_{\beta2} +0\,\bar g_{\beta3} +\ldots +1\,\bar g_{\beta,\partial_\beta-1} +\lambda_\beta\Phi\bar g_{\beta,\partial_\beta}, \end{aligned} \right. \quad (\beta=1,2,\ldots,r\le h) \tag{11*} \]

where the first equation of this system is equation \((1^*)\) for \(\lambda=\lambda_\beta\).

The system of \(\partial_\beta\) vector equations can also be represented in matrix-operator form as

\[ A[\bar g_{\beta1}\bar g_{\beta2}\cdots \bar g_{\beta\partial_\beta}] = \left[ \begin{array}{ccccc} \lambda_\beta\Phi & 0 & 0 & \cdots & 00\\ 1 & \lambda_\beta\Phi & 0 & \cdots & 00\\ 0 & \cdots & 1 & \lambda_\beta\Phi & \cdots 00\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & 1 & \lambda_\beta\Phi \end{array} \right] \left[ \begin{array}{c} \bar g_{\beta1}\\ \bar g_{\beta2}\\ \vdots\\ \vdots\\ \bar g_{\beta\partial_\beta} \end{array} \right] = \tag{11**} \]

\[ =\bigl[\bar g_{\beta 1}\bar g_{\beta 2}\cdots \bar g_{\beta\cdot \alpha-1}\cdots \bar g_{\beta\alpha}\cdots \bar g_{\beta\theta_\beta}\bigr] \begin{bmatrix} \lambda_\beta\Phi&1&0&\cdots&0&\cdots&0&0&\cdots&0\\ 0&\lambda_\beta\Phi&1&\cdots&0&0&0&&0\\ 0&0&0&\cdots&\lambda_\beta\Phi&1&\cdots&0&\cdots&0\\ 0&0&0&0&\lambda_\beta\Phi&1&&&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdots&0&0&0&\cdots&\lambda_\beta\Phi \end{bmatrix}, \]

to which the following matrix-vector relation corresponds:

\[ a_{(nn)} \bigl[g^\beta_{(n)1}\ g^\beta_{(n)2}\cdots g^\beta_{(n)\cdot \alpha-1}g^\beta_{(n)\alpha}\cdots g^\beta_{(n)\theta_\beta}\bigr] =a_{(nn)}g_{(n\theta_\beta)} \tag{11} \]

\[ =g_{(n\theta_\beta)} \begin{bmatrix} \lambda_\beta\varphi_{(nn)}&1&0&\cdots&0&0&\cdots&0\\ 0&\lambda_\beta\varphi_{(nn)}&1&\cdots&0&0&\cdots&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdots&\lambda_\beta\varphi_{(nn)}&1&\cdots&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdots&0&0&\cdots&\lambda_\beta\varphi_{(nn)} \end{bmatrix} =g_{(n\theta_\beta)}w_{(\theta_\beta\cdot\theta_\beta)}. \]

Thus, to each eigenvalue \(\lambda_\beta\) of multiplicity \(\theta_\beta\) of the operator pair \(A,\Phi\) there corresponds the matrix equality

\[ a_{(nn)}g_{(n\theta_\beta)}=g_{(n\theta_\beta)}w_{(\theta_\beta\cdot\theta_\beta)}, \tag{11a} \]

where \(w_{(\theta_\beta\theta_\beta)}\) is the generalized matrix

\[ w_{(\theta_\beta\theta_\beta)}= \begin{bmatrix} \lambda_\beta\varphi_{(nn)}&1&0&\cdots&0&0&\cdots&0\\ 0&\lambda_\beta\varphi_{(nn)}&1&\cdots&0&0&\cdots&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdots&\lambda_\beta\varphi_{(nn)}&1&\cdots&0\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&0&\cdots&0&0&\cdots&\lambda_\beta\varphi_{(nn)} \end{bmatrix}. \tag{12} \]

Now, instead of equality (7a)

\[ a_{(nn)}g_{(nr)}=\varphi_{(nn)}g_{(nr)}\lambda_{(rr)}, \tag{7a} \]

where

\[ g_{(nr)}=\bigl[g_{(n)1}g_{(n)2}\cdots g_{(n)\beta}\cdots g_{(n)r}\bigr],\quad (r\leq n) \tag{13} \]

and \(\lambda_{(rr)}\) is a purely diagonal matrix, we can compose the more general equality:

\[ a_{(nn)}g_{(nn)}=\varphi_{(nn)}g_{(nn)}w_{(nn)}, \tag{14} \]

in which

\[ 1)\quad g_{(nn)}=\bigl[g_{(n\theta_1)}g_{(n\theta_2)}\cdots g_{(n\theta_\beta)}\cdots g_{(n\theta_r)}\bigr], \tag{15} \]

\[ 2)\quad g_{(n\theta_\beta)}= \bigl[g^\beta_{(n)1}g^\beta_{(n)2}\cdots g^\beta_{(n)\alpha}\cdots g^\beta_{(n)\theta_\beta}\bigr], \tag{16} \]

\[ 3)\quad w_{(nn)}= \begin{bmatrix} w_{(\theta_1\theta_1)}&0_{(\theta_1\theta_2)}&\cdots&0_{(\theta_1\theta_\beta)}&\cdots&0_{(\theta_1\theta_r)}\\ 0_{(\theta_2\theta_1)}&w_{(\theta_2\theta_2)}& &0_{(\theta_2\theta_\beta)}&\cdots&0_{(\theta_2\theta_r)}\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0_{(\theta_\beta\theta_1)}&0_{(\theta_\beta\theta_2)}&\cdots&w_{(\theta_\beta\theta_\beta)}&\cdots&0_{(\theta_\beta\theta_r)}\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0_{(\theta_r\theta_1)}&0_{(\theta_r\theta_2)}&\cdots&0_{(\theta_r\theta_\beta)}&\cdots&w_{(\theta_r\theta_r)} \end{bmatrix}. \tag{17} \]

Thus, \(w_{(nn)}\) is a block-diagonal matrix. From equality (14) it follows that this matrix \(w_{(nn)}\) can be expressed directly in terms of the matrices \(a_{(nn)}\), \(\varphi_{(nn)}\), and \(g_{(nn)}\):

\[ w_{(nn)}=g_{(nn)}^{-1}\varphi_{(nn)}^{-1}a_{(nn)}g_{(nn)} =g_{(nn)}^{-1}b_{(nn)}g_{(nn)}, \tag{18} \]

\[ b_{(nn)}=\varphi_{(nn)}^{-1}a_{(nn)}. \tag{19} \]

Consideration of equality (18) with the notation (19) shows that, although according to (17) and (12) the matrix \(w_{(nn)}\) is represented in a generalized block-diagonal form, in reality \(w_{(nn)}\) is an ordinary block-diagonal matrix. What has been said follows from the fact that the equality (18) defining the matrix \(w_{(nn)}\) contains the ordinary \(nn\)-matrices \(a_{(nn)}\), \(g_{(nn)}\), \(\varphi_{(nn)}^{-1}\), and \(g_{(nn)}^{-1}\). The actual form of the blocks \(w_{(\partial_\beta\cdot\partial_\beta)}\) entering into \(w_{(nn)}\) will be indicated later.

1. The method set forth here for solving the generalized spectral problem (1) for a pair of matrices \(a_{(nn)}\), \(\varphi_{(nn)}\) of arbitrary form proves difficult for large \(n\). Namely, the computation of the coefficient \(b_s\) of the algebraic equation of degree \(n\), (6*), by expanding the corresponding determinant \(|a_{(nn)}-\lambda\varphi_{(nn)}|\) then becomes very difficult. Therefore in such cases it is expedient to construct directly the sought matrices \(g_{(nn)}\) and \(w_{(nn)}\), proceeding from equation (18) and applying the method of successive rotations.

Reducing equation (14) to the form (18) is equivalent to replacing the generalized spectral problem (1) for the matrices \(a_{(nn)}\), \(\varphi_{(nn)}\),

\[ a_{(nn)}g_{(n)}=\lambda\varphi_{(nn)}g_{(n)}. \tag{1} \]

by the ordinary spectral problem for the matrix

\[ b_{(nn)}=\varphi_{(nn)}^{-1}a_{(nn)}; \]

\[ b_{(nn)}g_{(n)}=\lambda g_{(n)}. \tag{1.0} \]

Therefore, for large \(n\), the generalized spectral problem (1), represented in the form (1.0), should expediently be solved by the mixed method of Danilevsky and successive rotations proposed by me, a detailed exposition of which is given in [2].

In the indicated method, the simplified-form matrix \(w_{(nn)}\), defined by the exact expression (18),

\[ w_{(nn)}=g_{(nn)}^{-1}b_{(nn)}g_{(nn)},\qquad b_{(nn)}=\varphi_{(nn)}^{-1}a_{(nn)}, \tag{18} \]

is obtained from the sequence of computational equalities

\[ w_{(nn)}^{(s+1)}= \left(g_{(nn)}^{(s)}\right)^{-1} w_{(nn)}^{(s)}g_{(nn)}^{(s)} =\sigma_{(nn)}^{(s)}b_{(nn)}^{(s)}g_{(nn)}^{(s)} =b_{(nn)}^{(s+1)},\qquad w_{(nn)}^{(0)}=b_{(nn)}, \tag{20} \]

which converge without bound to (18). Successive approximations to the eigenvalues \(\lambda_\beta\) and eigenvectors \(g_{(n)\beta}=\bar g_\beta\) of the matrix \(b_{(nn)}=\varphi_{(nn)}^{-1}a_{(nn)}\) are constructed by applying Danilevsky’s method to suitable diagonal blocks \(b_{(mm)}^{(s)}\) of the simplified matrices \(b_{(nn)}^{(s)}=w_{(nn)}^{(s)}\). In [2] the convergence of \(w_{(ss)}\) to \(w_{(nn)}\) as \(n\) increases is also proved.

In conclusion, we note that in [2] the following final form of the blocks \(w_{(\partial_\beta\cdot\partial_\beta)}\) entering into the composition of the matrix \(w_{(nn)}\) of the block-diagonal structure (17) was established directly.

\[ \mathfrak{w}_{(\partial_{\gamma};\partial_{\beta})} = \left[ \begin{array}{cccccccccc} \lambda_{\beta} & 1 & 0 & \cdots & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_{\beta} & 1 & \cdots & 0 & 0 & \cdots & 0 & 0 \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot & \cdots & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & \lambda_{\beta} & 1 & \cdots & 0 & 0 \\ \cdot & \cdot & \cdot & \cdots & \cdot & \cdot & \cdots & \cdot & \cdot \\ 0 & 0 & 0 & \cdots & 0 & 0 & \cdots & \lambda_{\beta} & 1 \\ 0 & 0 & 0 & \cdots & 0 & 0 & \cdots & 0 & \lambda_{\beta} \end{array} \right] \tag{21} \]

REFERENCES

1. B. F. Krutoi. Solution by the method of least squares of the spectral problem for the Helmholtz equation in the plane. The present collection.

2. B. F. Krutoi. Solution of the spectral problem for an arbitrary matrix \(a_{(nn)}\) by a generalized rotation method. The present collection.