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A GENERALIZATION OF PERRON’S THEOREM [1]
B. F. BYLOV
We shall write a system of homogeneous linear differential equations with real, bounded, and continuous coefficients on \(0 \le t < \infty\) in the form
\[ \frac{dx}{dt}=Ax, \tag{1} \]
where \(x\) is an \(n\)-dimensional vector and \(A=A(t)\) is a square matrix. To each ordered system of linearly independent solutions \(X=\{x_1,\ldots,x_n\}\) of equation (1) we assign the matrix \(X\), whose elements in the \(k\)-th column are the components of the vector \(x_k\). If the system is divided, without disturbing the order, into \(r\) groups, then we shall use the notation
\[ X=\left\{\frac{X_1}{n_1},\frac{X_2}{n_2},\ldots,\frac{X_r}{n_r}\right\}, \tag{2} \]
where \(n_k\) is the number of vectors belonging to the group \(X_k\), \(n_1+n_2+\cdots+n_r=n\), and \(X_k\) denotes a rectangular matrix of dimensions \(n\times n_k\) (in the sense indicated above). By the norm of a vector \(x\) we shall mean its Euclidean norm: \(\|x\|=\sqrt{(x,x)}\), and by the norm of a matrix—the maximum norm of the column vectors generating this matrix. Denote, respectively, by \(G(X)\) and \(G(X_k)\) the Gram determinants of the system of vectors \(X\) and of the groups \(X_k\) \((k=1,2,\ldots,r)\). Thus
\[ G(X)=\det X^*X=(\det X)^2 \quad \text{and} \quad G(X_k)=\det X_k^*X_k, \]
where \(^*\) denotes the operation of matrix transposition.
Theorem 1. If
\[ \inf_t \frac{G(X)}{G(X_1)G(X_2)\cdots G(X_r)}=\gamma>0, \tag{3} \]
then there exists a Lyapunov transformation \(x=Ly\) reducing (1) to the form
\[ \frac{dy}{dt}=By, \tag{4} \]
where \(B=\operatorname{diag}\{B_1,B_2,\ldots,B_r\}\) is a block-diagonal matrix, each block of which is an upper triangular matrix of order \(n_k\).
Proof. We orthonormalize separately each group \(X_k\) of vectors in the partition (2), using for this the well-known process in which the required vectors are found in the form of combinations:
\[ l_1=r_{11}x_1, \]
\[ l_2=r_{12}x_1+r_{22}x_2, \]
\[ \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot \]
\[ l_{n_1}=r_{1n_1}x_1+\cdots+r_{n_1n_1}x_{n_1}. \]
(The notation has been given only for the first group of vectors. For the remaining groups it is analogous.)
By means of linear algebra it is proved that, under the conditions
\[ (l_i,l_j)=\delta_{ij},\quad r_{ii}>0\quad (i,j=1,2,\ldots,n_1) \]
the coefficients \(r_{ij}\) in these combinations are determined uniquely. Orthonormalization of the group \(X_k\) is equivalent to multiplying the matrix \(X_k\) on the right by some upper triangular matrix \(R_k\) of order \(n_k\). Put \(R=\operatorname{diag}\{R_1,R_2,\ldots,R_r\}\) and \(L=XR\). We shall show that the transformation \(x=Ly\) is Lyapunov. For this, note first that, according to the construction, the norm of each column of the matrix \(L\) is equal to one, and therefore \(\det L\) is bounded above. Further using the condition of the theorem and the block structure of \(R\), we have
\[ (\det L)^2=(\det R)^2G(X)\ge \gamma(\det R)^2G(X_1)G(X_2)\cdots G(X_r)= \]
\[ =\gamma(\det R_1)^2G(X_1)(\det R_2)^2G(X_2)\cdots(\det R_r)^2G(X_r)= \]
\[ =\gamma\det(R_1^{*}X_1^{*}X_1R_1)\cdot\det(R_2^{*}X_2^{*}X_2R_2)\cdots\det(R_r^{*}X_r^{*}X_rR_r). \]
Each factor \(\det(R_k^{*}X_k^{*}X_kR_k)\) on the right-hand side of the preceding relation is the Gram determinant of the collection of orthonormal vectors \(X_kR_k\) and therefore is equal to 1. Consequently, \((\det L)^2\ge\gamma>0\), which proves the existence and boundedness of the matrix \(L^{-1}\). To prove the boundedness of \(L'\), represent \(L\) in the form
\[ L=\{L_1,L_2,\ldots,L_r\},\quad \text{where } L_k=X_kR_k. \]
Fixing some \(k\), consider the system of vectors
\[ \bar X=\{X_k,X_1,\ldots,X_{k-1},X_{k+1},\ldots,X_r\}, \]
which differs from \(X\) only in the order of writing.
Starting from the matrix \(\bar X\), we find the matrix \(Q\) of a Perron transformation \(x=Qz\) reducing equation (1) to triangular form. In this case \(Q=\bar X\bar R\), where \(\bar R\) is an upper triangular matrix orthonormalizing the system of vectors \(\bar X\) [2].
Since the system of vectors \(\bar X\) begins with the group \(X_k\), the first \(n_k\) columns of the matrix \(Q\) will then be obtained by orthonormalizing the vectors of the group \(X_k\), and therefore will coincide with the corresponding columns of the matrix \(L_k=X_kR_k\). Since the matrix of the Perron transformation has a bounded derivative, the matrix \(L_k'\) is bounded. The preceding arguments are applicable for any \(k=1,2,\ldots,r\), and thus the boundedness of \(L'\) is established.
Finally, let us show that the matrix \(B\) in equation (4), obtained from (1) by the transformation \(x=Ly\), has the required structure. Let, under this transformation, to the system of solutions \(X\) of equation (1) there correspond some system of solutions \(Y\) of equation (4).
Then
\[ Y=L^{-1}X=(XR)^{-1}X=R^{-1}=\operatorname{diag}\{R_1^{-1},\ R_2^{-1},\ \ldots,\ R_r^{-1}\}. \]
In other words, equation (4) has a system of independent solutions \(Y\) determined by a block-diagonal matrix, each block \(R_k^{-1}\) of which is an upper triangular matrix. But this, obviously, can occur only in the case when the matrix \(B\) in equation (4) itself has the block-diagonal structure described in the assertion of the theorem, etc.
The converse is true.
Theorem 2. If equation (1) can be brought by some Lyapunov transformation to the form (4), then equation (1) has a system of solutions \(X\) for which (3) is satisfied.
Proof. Let the transformation \(x=Ly\) bring (1) to the form (4), where \(B\) has a block-diagonal structure. Let us find a system of solutions \(Y\) of equation (4) for which \(Y(0)=E_n\) (\(E_n\) is the identity matrix of order \(n\)).
Equation (4) decomposes into \(r\) vector equations
\[ \frac{d\bar y}{dt}=B_k\bar y\quad (k=1,\ 2,\ \ldots,\ r), \]
which can be integrated independently of one another. Having found a system of solutions \(\bar Y_k\) \((\bar Y_k(0)=E_{n_k})\) of each of these equations, we shall have \(Y=\operatorname{diag}\{\bar Y_1,\ \bar Y_2,\ \ldots,\ \bar Y_r\}\).
We divide this system of solutions into \(r\) groups \(Y=\{Y_1,\ Y_2,\ \ldots,\ Y_r\}\), where each rectangular matrix \(Y_k\) (of size \(n\times n_k\)) is generated by the square matrix \(\bar Y_k\), “above” and “below” which a certain number of rows consisting of zeros are written.
We likewise divide the Lyapunov transformation matrix into \(r\) rectangular matrices \(L=\{L_1,\ L_2,\ \ldots,\ L_r\}\).
Let us find the system of solutions \(X=LY\) of equation (1), corresponding, by virtue of the Lyapunov transformation, to the system of solutions \(Y\) of equation (4). In this case the group of solutions \(Y_k\) from \(Y\) corresponds to the group of solutions \(X_k=LY_k\) from \(X\). Thus \(X\) is also divided into \(r\) groups: \(X=\{X_1,\ X_2,\ \ldots,\ X_r\}\). Taking into account the structure of the matrix \(Y_k\), we note that
\[ X_k=LY_k=L_k\bar Y_k. \]
Using Liouville’s formula, we further find
\[ G(X)=(\det X)^2=(\det L)^2\cdot(\det Y)^2 =G(L)\exp 2\int_0^t \operatorname{sp} B\,d\tau, \]
\[ G(X_k)=\det X_k^{*}X_k=\det \bar Y_k^{*}L_k^{*}L_k\bar Y_k \]
\[ =(\det \bar Y_k)^2G(L_k) =G(L_k)\exp 2\int_0^t \operatorname{sp} B_k\,d\tau. \]
Therefore
\[ \frac{G(X)}{G(X_1)G(X_2)\ldots G(X_r)} = \frac{G(L)}{G(L_1)G(L_2)\ldots G(L_r)}. \]
By the properties of the Lyapunov transformation, the right-hand side of the equality exceeds a positive constant, as was required to prove.
Corollary. For reducibility by a Lyapunov transformation \(x=Ly\) of equation (1) to the diagonal form \(\dfrac{dy}{dt}=By\), where \(B=\operatorname{diag}\{b_{11}, b_{22}, \ldots, b_{nn}\}\), it is necessary and sufficient that equation (1) have a system of solutions \(X=\{x_1, x_2, \ldots, x_n\}\) for which
\[ \inf_t \frac{G(X)}{\|x_1\|^2\cdot \|x_2\|^2 \cdots \|x_n\|^2}=\gamma>0. \]
References
- Perron O. Math. Zeitschr., 32, 1930.
- Vinograd R. E. Uspekhi Mat. Nauk, vol. IX, no. 2 (60), 129–136, 1954.
Received by Uspekhi Mat. Nauk
September 31, 1963.
Moscow Aviation
Technological Institute