Abstract
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MATHEMATICS
D. F. DAVIDENKO
ON THE APPLICATION OF THE METHOD OF VARIATION OF A PARAMETER TO THE COMPUTATION OF DETERMINANTS
(Presented by Academician N. N. Bogolyubov, 16 XI 1959)
1°. Let there be given a square matrix \(A(\lambda)=\|a_{ij}(\lambda)\|\) \((i,j=1,2,\ldots,n)\) of order \(n\), whose elements are functions of a parameter \(\lambda\), taking prescribed values on some finite interval \(\lambda_0 \leqslant \lambda \leqslant \lambda^*\). We shall call the matrix \(A^{-1}(\lambda)=\|\alpha_{ij}(\lambda)\|\) \((i,j=1,2,\ldots,n)\) the inverse of the matrix \(A(\lambda)\), if
\[ A^{-1}(\lambda)A(\lambda)=E, \tag{1} \]
where \(E\) is the identity matrix.
Suppose that the functions \(a_{ij}(\lambda)\) \((i,j=1,2,\ldots,n)\) are defined and continuous on the whole interval \(\lambda_0 \leqslant \lambda \leqslant \lambda^*\), and have continuous derivatives on this interval. Suppose, moreover, that the matrix \(A(\lambda)\) has on the interval \(\lambda_0 \leqslant \lambda \leqslant \lambda^*\) a determinant \(\Delta(\lambda)\) different from zero, and that for some value of \(\lambda\), for example \(\lambda=\lambda_0\), the value of this determinant is known to us:
\[ \Delta(\lambda_0)=\Delta^{(0)}. \tag{2} \]
It is required to find approximate values of \(\Delta(\lambda)\) for prescribed values \(\lambda>\lambda_0\).
Denote by \(dA(\lambda)/d\lambda=\|a'_{ij}(\lambda)\|\) the matrix obtained from the matrix \(A(\lambda)\) by replacing all its elements by their derivatives with respect to \(\lambda\), and prove the following lemma.
Lemma. If for the matrix \(A(\lambda)=\|a_{ij}(\lambda)\|\) \((i,j=1,2,\ldots,n)\) there exists on the interval \(\lambda_0 \leqslant \lambda \leqslant \lambda^*\) the inverse matrix \(A^{-1}(\lambda)=\|\alpha_{ij}(\lambda)\|\) \((i,j,\ldots,n)\), then for all \(\lambda\) from this interval the relation holds*
\[ \frac{d\Delta(\lambda)}{d\lambda} = \Delta(\lambda)\operatorname{Sp}\left(A^{-1}(\lambda)\frac{dA(\lambda)}{d\lambda}\right)^{**}. \tag{3} \]
Proof. Taking \(\lambda\) as the independent variable and differentiating with respect to \(\lambda\) the determinant
\[ \Delta(\lambda)= \begin{vmatrix} a_{11}(\lambda) & a_{12}(\lambda) & \ldots & a_{1n}(\lambda)\\ a_{21}(\lambda) & a_{22}(\lambda) & \ldots & a_{2n}(\lambda)\\ \cdot & \cdot & \cdot & \cdot\\ a_{n1}(\lambda) & a_{n2}(\lambda) & \ldots & a_{nn}(\lambda) \end{vmatrix}, \]
* An analogous lemma was proved in a somewhat different way in (1).
** Here \(\operatorname{Sp} D\) denotes the trace of the matrix \(D\), i.e. the sum of its diagonal elements.
we find
\[ \frac{d\Delta(\lambda)}{d\lambda} = \sum_{l=1}^{n} \left| \begin{array}{ccccccc} a_{11}(\lambda)&\ldots&a_{1,l-1}(\lambda)&a'_{1l}(\lambda)&a_{1,l+1}(\lambda)&\ldots&a_{1n}(\lambda)\\ a_{21}(\lambda)&\ldots&a_{2,l-1}(\lambda)&a'_{2l}(\lambda)&a_{2,l+1}(\lambda)&\ldots&a_{2n}(\lambda)\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ a_{n1}(\lambda)&\ldots&a_{n,l-1}(\lambda)&a'_{nl}(\lambda)&a_{n,l+1}(\lambda)&\ldots&a_{nn}(\lambda) \end{array} \right|. \tag{4} \]
Further, differentiating identity (1) with respect to \(\lambda\), we have
\[ \frac{dA^{-1}(\lambda)}{d\lambda} A(\lambda) + A^{-1}(\lambda)\frac{dA(\lambda)}{d\lambda} =0, \tag{5} \]
whence
\[ \frac{dA(\lambda)}{d\lambda} = - A(\lambda)\frac{dA^{-1}(\lambda)}{d\lambda}A(\lambda), \]
or, in expanded form,
\[ a'_{ij}(\lambda) = - \sum_{\nu,\mu=1}^{n} a_{i\nu}(\lambda)\alpha'_{\nu\mu}(\lambda)a_{\mu j}(\lambda) \quad (i,\ j=1,\ 2,\ldots,n). \tag{6} \]
Substituting (6) into (4), we obtain
\[ \frac{d\Delta(\lambda)}{d\lambda} = -\Delta(\lambda)\sum_{p,q=1}^{n}\alpha'_{pq}(\lambda)a_{qp}(\lambda), \tag{7} \]
where, evidently,
\[ \sum_{p,q=1}^{n}\alpha'_{pq}(\lambda)a_{qp}(\lambda) = \operatorname{Sp}\left(\frac{dA^{-1}(\lambda)}{d\lambda}A(\lambda)\right). \]
Taking identity (5) into account, we arrive at the required relation (3). The lemma is proved.
To determine, for given \(\lambda\), the values of the determinant \(\Delta(\lambda)\), we numerically integrate the differential equation (3) on the interval \(\lambda_0 \leq \lambda \leq \lambda^*\) under the initial condition (2):
\[ \lambda=\lambda_0,\qquad \Delta(\lambda)=\Delta^0. \]
The numerical values \(\Delta(\lambda)\) obtained in the integration for each specified value of the parameter \(\lambda\) will be the desired approximate values of the determinant of the matrix \(A(\lambda)\).
We note that, in the numerical integration of equation (3), it is necessary to know at each step the elements of the inverse matrix \(A^{-1}(\lambda)\). In computing these elements one may use, for example, the method of parameter variation \((^2)\). In doing so one should additionally assume that for \(\lambda=\lambda_0\) the inverse matrix \(A^{-1}(\lambda)\) is known:
\[ A^{-1}(\lambda_0)=A_0^{-1}=\|\alpha_{ij}^{(0)}\| \quad (i,\ j=1,\ 2,\ldots,n), \tag{8} \]
and, instead of one equation (2), numerically integrate the system of two equations
\[ \begin{aligned} \frac{d\Delta(\lambda)}{d\lambda} &= \Delta(\lambda)\operatorname{Sp}\left(A^{-1}(\lambda)\frac{dA(\lambda)}{d\lambda}\right),\\ \frac{dA^{-1}(\lambda)}{d\lambda} &= - A^{-1}(\lambda)\frac{dA(\lambda)}{d\lambda}A^{-1}(\lambda) \end{aligned} \tag{9} \]
under the initial conditions (2), (8):
\[ \lambda=\lambda_0,\qquad \Delta(\lambda)=\Delta^{(0)},\qquad A^{-1}(\lambda)=A_0^{-1}. \]
\(2^\circ.\) Let now, for some value \(\lambda=\lambda_i\) from the interval
for \(\lambda_0 \ll \lambda \ll \lambda^*\) the determinant \(\Delta(\lambda)\) vanishes, i.e.,
\[ \Delta(\lambda_i)=0. \]
Let us write the matrix \(A(\lambda)\) in the form
\[ A(\lambda)= \begin{Vmatrix} P(\lambda) & u(\lambda)\\ v(\lambda) & a_{nn}(\lambda) \end{Vmatrix}, \]
where \(P(\lambda)\) is a matrix of order \((n-1)\), \(v(\lambda)\) is a row, \(u(\lambda)\) is a column, and suppose that the determinant \(\overline{\Delta}(\lambda)\) of the matrix \(P(\lambda)\) in some neighborhood of the point \(\lambda_i\), say \(\lambda_i' \ll \lambda \ll \lambda_i''\), is nonzero.
We write the inverse matrix \(A^{-1}(\lambda)\), according to the bordering method \({}^{(3)}\), in the form
\[ A^{-1}(\lambda)= \begin{Vmatrix} P^{-1}(\lambda)+\dfrac{P^{-1}(\lambda)u(\lambda)v(\lambda)P^{-1}(\lambda)}{\alpha(\lambda)} & -\dfrac{P^{-1}(\lambda)u(\lambda)}{\alpha(\lambda)} \\[1.2em] -\dfrac{v(\lambda)P^{-1}(\lambda)}{\alpha(\lambda)} & \dfrac{1}{\alpha(\lambda)} \end{Vmatrix}, \tag{10} \]
where \(P^{-1}(\lambda)\) is the inverse matrix to the matrix \(P(\lambda)\),
\[ \alpha(\lambda)=a_{nn}(\lambda)-v(\lambda)P^{-1}(\lambda)u(\lambda). \]
The relation \({}^{(4,5)}\) holds
\[ \alpha(\lambda)=\frac{\Delta(\lambda)}{\overline{\Delta}(\lambda)}, \tag{11} \]
from which it is seen that \(\Delta(\lambda)\) and \(\alpha(\lambda)\) vanish simultaneously.
Thus, in order to find the numerical values of \(\Delta(\lambda)\) in a neighborhood of the point \(\lambda_i\), on the interval \(\lambda_i' \ll \lambda \ll \lambda_i''\), instead of the system (9) we numerically integrate the system
\[ \frac{d\Delta(\lambda)}{d\lambda} = \operatorname{Sp}\left(B(\lambda)\frac{dA(\lambda)}{d\lambda}\right), \]
\[ B(\lambda) = \overline{\Delta}(\lambda) \begin{Vmatrix} \alpha(\lambda)P^{-1}(\lambda)+M(\lambda) & -P^{-1}(\lambda)u(\lambda)\\ -v(\lambda)P^{-1}(\lambda) & 1 \end{Vmatrix}, \]
\[ M(\lambda)=P^{-1}(\lambda)u(\lambda)v(\lambda)P^{-1}(\lambda), \]
\[ \alpha(\lambda)=a_{nn}(\lambda)-v(\lambda)P^{-1}(\lambda)u(\lambda), \tag{12} \]
\[ \frac{dP^{-1}(\lambda)}{d\lambda} = -P^{-1}(\lambda)\frac{dP(\lambda)}{d\lambda}P^{-1}(\lambda), \]
\[ \frac{d\overline{\Delta}(\lambda)}{d\lambda} = \overline{\Delta}(\lambda)\operatorname{Sp}\left(P^{-1}(\lambda)\frac{dP(\lambda)}{d\lambda}\right), \]
where \(B(\lambda)\) is the adjugate matrix for the matrix \(A(\lambda)\), obtained from (10) with the aid of relation (11), with initial conditions
\[ \lambda=\lambda_i',\qquad \Delta(\lambda)=\Delta',\qquad P^{-1}(\lambda)=P_0^{-1},\qquad \overline{\Delta}(\lambda)=\overline{\Delta}'. \]
Here, as the initial value \(\Delta'\) for the determinant \(\Delta(\lambda)\) we take its last value \(\Delta(\lambda_i')\), obtained in integrating system (9). The initial matrix \(P_0^{-1}\) and the value \(\overline{\Delta}'\) of the determinant \(\overline{\Delta}(\lambda)\) are found from the formulas
\[ P_0^{-1}=Q(\lambda_i')-\frac{r(\lambda_i')q(\lambda_i')}{a_{nn}(\lambda_i')}, \qquad \overline{\Delta}'=\alpha_{nn}(\lambda_i')\Delta(\lambda_i'). \]
in which the elements of the inverse matrix are used
\[ A^{-1}(\lambda_i')= \left\| \begin{matrix} Q(\lambda_i') & r(\lambda_i')\\ q(\lambda_i') & \alpha_{nn}(\lambda_i') \end{matrix} \right\|, \]
obtained by integrating system (9).
Here \(Q(\lambda_i')\) is a matrix, \(q(\lambda_i')\) a row, \(r(\lambda_i')\) a column, and \(\alpha_{nn}(\lambda_i')\) a number.
After system (12) has been integrated on the interval \(\lambda_i' \leq \lambda \leq \lambda_i''\), we again return to system (9) and continue to integrate it numerically on the interval \(\lambda_i'' \leq \lambda \leq \lambda^*\), taking as the initial value for \(\Delta(\lambda)\) its last value \(\Delta(\lambda_i'')\), obtained when integrating system (12), and determining the initial matrix for \(A^{-1}(\lambda)\) by formula (10) for \(\lambda=\lambda_i''\).
3°. The proposed method may also be applied to the computation of determinants of constant matrices. In this case the constant \(n\)-dimensional matrix \(C\) is represented in the form of the sum of two matrices \(C_0\) and \(C_1\) so that the matrix \(C_0^{-1}\) and the value of the determinant \(|C_0|\) can be easily found. Then the matrix
\[ C_\lambda=C_0+\lambda C_1 \]
for \(\lambda=1\) coincides with the original matrix \(C\), while for \(\lambda=0\) it has the inverse \(C_0^{-1}\) and the determinant value \(|C_\lambda|\) equal to the determinant value \(|C_0|\).
Proceeding with the matrix \(C_\lambda\) analogously to what was set out in 1°, we obtain for the values of the determinant \(|C_\lambda|\) the following system of equations, analogous to system (9):
\[ \frac{d|C_\lambda|}{d\lambda}=|C_\lambda|\operatorname{Sp}(C_\lambda^{-1}C_1), \]
\[ \frac{dC_\lambda^{-1}}{d\lambda}=-C_\lambda^{-1}C_1C_\lambda^{-1}. \]
We integrate this system numerically on the interval \(0 \leq \lambda \leq 1\). As the initial conditions we take the values of \(|C_\lambda|\) and \(C_\lambda^{-1}\) corresponding to \(\lambda=0\). The value of the determinant \(|C_\lambda|\) for \(\lambda=1\) will be the desired value of the determinant of the matrix \(C\).
Received
22 X 1959
CITED LITERATURE
- A. Khalanai, On linear differential equations of the 2nd order with almost-periodic coefficients, Dissertation, 1952.
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- V. N. Faddeeva, Computational Methods of Linear Algebra, M.—L., 1950.
- N. F. Kovaleva, L. S. Mayants, DAN, 108, No. 2 (1956).
- A. Raḥman, Bull. Cl. Sci. Acad. Roy. Belg., 40, No. 8, 798 (1954).