Reports of the Academy of Sciences of the USSR

1. Vol. 131, No. 5

MATHEMATICS

D. F. DAVIDENKO

ON THE APPLICATION OF THE METHOD OF VARIATION OF A PARAMETER TO THE COMPUTATION OF EIGENVALUES AND EIGENVECTORS OF MATRICES

(Presented by Academician N. N. Bogolyubov on 16 XI 1959)

1°. Let there be given a square matrix (A(\lambda)=|a_{kj}(\lambda)|) ((k,j=1,2,\ldots,n)) of order (n), whose elements are functions of the parameter (\lambda), taking prescribed values on some finite interval (\lambda_0 \leq \lambda \leq \lambda^*).

By the eigenvalues of the matrix (A(\lambda)) we shall, as usual, mean the solutions (p_i=p_i(\lambda)) ((i=1,2,\ldots,n)) of its characteristic equation

[
\operatorname{Det}|A(\lambda)-pE|=0.
]

To each eigenvalue (p_i(\lambda)) we assign a nonzero eigenvector (X_i=X_i(\lambda)) satisfying the equation

[
A(\lambda)X_i=p_iX_i.
]

In what follows we shall denote by (B(\lambda,p_i)=|b_{kj}(\lambda,p_i)|) ((k,j=1,2,\ldots,n)) the matrix (|A(\lambda)-p_iE|), and by (\omega(\lambda,p_i)) its determinant, i.e.

[
\omega(\lambda,p_i)=\operatorname{Det} B(\lambda,p_i)=0.
\tag{1}
]

By (C(\lambda,p_i)=|c_{kj}(\lambda,p_i)|) ((k,j=1,2,\ldots,n)), where (c_{kj}(\lambda,p_i)) is the cofactor of the element (b_{jk}(\lambda,p_i)) in the determinant (\omega(\lambda,p_i)), we shall denote the adjugate matrix for the matrix (B(\lambda,p_i)).

Suppose that for some prescribed value of the parameter (\lambda) from the interval (\lambda_0 \leq \lambda \leq \lambda^*), say (\lambda=\lambda_0), the value of the eigenvalue (p_i(\lambda)) is known to us:

[
\text{for } \lambda=\lambda_0 \qquad p_i(\lambda)=p_i^{(0)}.
\tag{2}
]

Suppose, moreover, that:

1) all the functions (a_{kj}(\lambda)) are defined and continuous on the whole interval (\lambda_0 \leq \lambda \leq \lambda^*) and have continuous derivatives on this interval;

2) the trace of the matrix (C(\lambda,p_i)\dfrac{\partial B(\lambda,p_i)}{\partial p_i}),

[
\operatorname{Sp}\left[
C(\lambda,p_i)\frac{\partial B(\lambda,p_i)}{\partial p_i}
\right]
]

at the point ((\lambda_0,p_i^{(0)})) is different from zero.

It is required to find approximate values of the eigenvalue (p_i(\lambda)) for prescribed values of the parameter (\lambda > \lambda_0).

For this purpose we proceed as follows. Differentiating equation (1) with respect to (\lambda) and then solving it with respect to (dp_i/d\lambda), we obtain

[
\frac{dp_i}{d\lambda}
=
-
\frac{\partial \omega(\lambda,p_i)/\partial \lambda}
{\partial \omega(\lambda,p_i)/\partial p_i}.
]

By virtue of the lemma proved in ((^{1})), the last equation can be rewritten in the form (cf. ((^{2})))

[
\frac{dp_i}{d\lambda}
=
-
\frac{
\operatorname{Sp}\left[
C(\lambda,p_i)\,\frac{\partial B(\lambda,p_i)}{\partial \lambda}
\right]
}{
\operatorname{Sp}\left[
C(\lambda,p_i)\,\frac{\partial B(\lambda,p_i)}{\partial p_i}
\right]
}.
\tag{3}
]

Let the trace of the matrix

[
C(\lambda,p_i)\,\frac{\partial B(\lambda,p_i)}{\partial p_i}
]

be different from zero at all points of the domain (G^{(i)}) of variation of (\lambda) and (p_i), containing the point ((\lambda_0,p_i^{(0)})), i.e.

[
\operatorname{Sp}\left[
C(\lambda,p_i)\,\frac{\partial B(\lambda,p_i)}{\partial p_i}
\right]
\ne 0
\quad \text{in } G^{(i)}.
\tag{4}
]

Then, in order to determine, for prescribed (\lambda), the solutions of equation (1), or, what is the same thing, the eigenvalue values (p_i(\lambda)) of the matrix (A(\lambda)), we numerically integrate equation (3) on the interval (\lambda_0 \leq \lambda \leq \lambda^*) with the initial condition (2).

When computing the elements of the adjugate matrix (C(\lambda,p_i)), we proceed as follows. Represent the matrix (B(\lambda,p_i)) in the form

[
B(\lambda,p_i)
=
\left|
\begin{array}{cc}
P(\lambda,p_i) & u(\lambda,p_i)\
v(\lambda,p_i) & b_{n,n}(\lambda,p_i)
\end{array}
\right|,
]

where (P(\lambda,p_i)) is a matrix of order (n-1) with determinant (\overline{\Delta}(\lambda,p_i)) different from zero in the domain (G^{(i)});

[
v(\lambda,p_i)={b_{n,1}(\lambda,p_i),\ldots,b_{n,n-1}(\lambda,p_i)},
\qquad
u(\lambda,p_i)=
\begin{pmatrix}
b_{1,n}(\lambda,p_i)\
\cdot\
\cdot\
b_{n-1,n}(\lambda,p_i)
\end{pmatrix}.
]

In the case when the determinant (\overline{\Delta}(\lambda,p_i)) vanishes at some point of the domain (G^{(i)}), the matter is reduced to the case under consideration by a corresponding interchange of rows and columns of the matrix (B(\lambda,p_i)).

Proceeding now analogously to the way we did in ((^{1})), the adjugate matrix (C(\lambda,p_i)) can be represented in the form

[
C(\lambda,p_i)
=
\overline{\Delta}(\lambda,p_i)\,C^*(\lambda,p_i),
]

where

[
C^*(\lambda,p_i)
=
\left|
\begin{array}{cc}
F(\lambda,p_i) & -P^{-1}(\lambda,p_i)u(\lambda,p_i)\
-v(\lambda,p_i)P^{-1}(\lambda,p_i) & 1
\end{array}
\right|,
]

[
F(\lambda,p_i)
=
P^{-1}(\lambda,p_i)u(\lambda,p_i)v(\lambda,p_i)P^{-1}(\lambda,p_i).
]

The computation of the values of the elements of the inverse matrix (P^{-1}(\lambda,p_i)) may be carried out, for example, by the method of variation of the parameter ((^3)). For this it is necessary additionally to assume that, under condition (2), the inverse matrix (P^{-1}(\lambda,p_i)) is known to us:

[
\text{for } \lambda=\lambda_0 \qquad p_i(\lambda)=p_i^{(0)}, \qquad
P^{-1}(\lambda,p_i)=P_0^{-1}.
\tag{5}
]

Thus, the entire computational process reduces to the numerical integration, on the interval (\lambda_0 \leqslant \lambda \leqslant \lambda^*), of the system of two equations

[
\frac{dp_i}{d\lambda}
=
-
\frac{
\operatorname{Sp}\left[
C^(\lambda,p_i)\dfrac{\partial B(\lambda,p_i)}{\partial \lambda}
\right]
}{
\operatorname{Sp}\left[
C^(\lambda,p_i)\dfrac{\partial B(\lambda,p_i)}{\partial p_i}
\right]
},
\tag{6}
]

[
\frac{dP^{-1}(\lambda,p_i)}{d\lambda}
=
-
P^{-1}(\lambda,p_i)
\frac{dP(\lambda,p_i)}{d\lambda}
P^{-1}(\lambda,p_i).
]

with the initial conditions (5).

It is easy to show that in the domain (G^{(i)}) each column of the matrix (C^*(\lambda,p_i)) consists of components of the eigenvector (X_i) belonging to the eigenvalue (p_i(\lambda)).

Above we assumed that condition (4) is satisfied. Let us note that if all eigenvalues (p_1(\lambda), p_2(\lambda),\ldots,p_n(\lambda)) of the matrix (A(\lambda)) take distinct values on the interval (\lambda_0 \leqslant \lambda \leqslant \lambda^*), then condition (4) will be satisfied on this interval for any (i=1,2,\ldots,n).

Let us also note that of particular interest are the cases when, at some point of the domain (G^{(i)}) which is a solution of equation (1), the trace of the matrix

[
C(\lambda,p_i)\frac{\partial B(\lambda,p_i)}{\partial p_i}
]

vanishes. In these cases we proceed analogously to how we did in the corresponding cases in (4).

(2^\circ). The proposed method is also applicable to the computation of eigenvalues and eigenvectors of constant matrices. In this case the constant matrix (D) is represented as the sum of two matrices (D_0) and (D_1) in such a way that the eigenvalues of the matrix (D_0) and the matrix (P_0^{-1}) for the matrix (|D_0-pE|) are easily determined. Then the matrix (D_\lambda=D_0+\lambda D_1) for (\lambda=1) coincides with the original matrix (D), while for (\lambda=0) its eigenvalues and the matrix (P_0^{-1}) are known.

We treat the matrix (D_\lambda) analogously to what was set out in (1^\circ), and integrate the obtained system of differential equations of the form (6) on the interval (0 \leqslant \lambda \leqslant 1). The desired values of the eigenvalues and the components of the eigenvectors are obtained at (\lambda=1).

(3^\circ). Example. Suppose it is required to find the eigenvalues and the corresponding eigenvectors of the matrix

[
A(\lambda)=
\begin{Vmatrix}
0 & \lambda & \lambda & 2\lambda-1\
\lambda & -\lambda & \lambda & \lambda\
\lambda & \lambda-1 & \lambda & \lambda\
2\lambda-1 & \lambda & \lambda & -1
\end{Vmatrix}
\tag{7}
]

for the following values of the parameter (\lambda): (0;\ 0.05;\ 0.1;\ldots;\ 0.95;\ 1).

For (\lambda=0) the eigenvalues of matrix (7) take the values:
(p_1^{(0)}=0;\ p_2^{(0)}=0.618034;\ p_3^{(0)}=-1;\ p_4^{(0)}=-1.618034.)

Choosing the step of numerical integration of the system (6) equal to (0.05) up to (\lambda=0.25), and then equal to (0.025), and using the Runge--Kutta method,

we obtain for the eigenvalue (p_1(\lambda)) of matrix (7) the following results

For the components of the corresponding eigenvector (X_1) at (\lambda=1) we obtain the values: (-2.302858;\ 1.000055;\ 1.000060;\ 1). The residual vector: (-0.000023;\ 0.000008;\ 0.000005;\ 0.000046).

Note added in proof. After the present note had been submitted for publication, the author became aware of a paper by A. A. Dorodnitsyn (DAN, 126, No. 6, 1959), in which differential equations were also obtained for determining the eigenvalues and eigenvectors of the symmetric matrix (C=A+\varepsilon B) ((A, B) constant matrices, (\varepsilon) a parameter).

Received
22 X 1959

CITED LITERATURE

1. D. F. Davidenko, DAN, 131, No. 4 (1959).
2. I. F. Kovalev, L. S. Mayants, DAN, 108, No. 2 (1956).
3. D. F. Davidenko, DAN, 131, No. 3 (1959).
4. D. F. Davidenko, Ukr. Mat. Zh., 5, No. 2 (1953).