UDC 517.941.1

MATHEMATICS

S. M. LOZINSKII

ESTIMATE OF THE DIFFERENCE OF TWO MATRICANTS

(Presented by Academician V. I. Smirnov on 25 IV 1966)

1. Notation. The word matrix means a square matrix of fixed order \(n\), whose elements are complex numbers. If \(A\) is a matrix, then \([A]_{\mu\nu}\) \((\mu,\nu=1,\ldots,n)\) are its elements; \(|A|\) denotes the matrix whose elements are the moduli of the corresponding elements of \(A\); \(A^{+}\) denotes the matrix whose diagonal (off-diagonal) elements are the real parts (moduli) of the corresponding elements of \(A\); \(|A|\) is called the modulus of the matrix \(A\). The symbol \(E\) denotes the identity matrix; the symbol \(I\) denotes the matrix all of whose elements are equal to one. A matrix-function is a matrix whose elements are functions of the real parameter \(t\). If \(A=A(t)\) is a matrix-function, then
\[ A^{+}(t)\stackrel{\mathrm{def}}{=}(A(t))^{+}. \]
Inequalities between matrices, convergence of a sequence of matrices and of a matrix series, and also continuity and differentiability of matrix-functions are understood elementwise. The letters \(A,B,C,D,P,Q\) (also with indices) denote matrices or, if this is stipulated, matrix-functions; the latter are always assumed to be continuous. It is useful to note that
\[ |A+B|\leq |A|+|B|,\qquad |AB|\leq |A|\,|B|. \]

A norm is introduced in the set of matrices, satisfying the three axioms of the norm of a linear complex space and the conditions
\[ \|AB\|\leq \|A\|\,\|B\|,\qquad \|E\|=1. \]
It is known that
\[ \lim_{k\to\infty}\|A_k-A\|=0 \]
is equivalent to
\[ \lim_{k\to\infty} A_k=A \]
(see \((^{1})\), Chap. 1, § 13, no. 3). The norm is used below only in the formulation and proof of Theorem 2 (paras. 4, 8, 10; see also paras. 12, 13).

2. Consider the linear homogeneous system of differential equations (in vector notation)
\[ dx/dt=A(t)x, \tag{1} \]
where \(A(t)\) is a matrix-function on the interval \([t_0,T]\). The symbol
\[ {}_{\xi}^{t}\Omega A \]
(where \(t_0\leq \xi,\ t\leq T\)) denotes the matricant of the matrix-function \(A(t)\), i.e. the matrix-function whose columns are the solutions of system (1) satisfying the condition
\[ {}_{\xi}^{\xi}\Omega A=E \]
(for the properties of the matricant, see \((^{2})\), Chap. XIV, §§ 5, 6). If, along with system (1), the system \(dx/dt=B(t)x\) is also considered, then it is often necessary to estimate
\[ {}_{t_0}^{t}\Omega B-{}_{t_0}^{t}\Omega A. \]

Formulation of the results.

3. Theorem 1 (estimate of the modulus of the difference of matricants). Let \(A,B,C,D\) be matrix-functions on \([t_0,T]\), and let
\[ A^{+}(t)\leq C(t),\qquad |B(t)-A(t)|\leq D(t)\quad \text{for } t_0\leq t\leq T. \tag{2} \]
Then:

a)
\[ \left|{}_{t_0}^{t}\Omega B-{}_{t_0}^{t}\Omega A\right| \leq {}_{t_0}^{t}\Omega(C+D)-{}_{t_0}^{t}\Omega C \quad \text{for } t_0\leq t\leq T. \]

b) If, moreover, the matrices \(C(t_1), C(t_2), D(t_1), D(t_2)\) commute pairwise for \(t_0 \leqslant t_1, t_2 \leqslant T\), then

\[ \left| \Omega_{t_0}^{t} B-\Omega_{t_0}^{t} A \right| \leqslant \exp\int_{t_0}^{t} C(u)\,du \cdot \left\{ \exp\int_{t_0}^{t} D(u)\,du-E \right\} \quad \text{for } t_0 \leqslant t \leqslant T . \]

c) In particular, if

\[ C(t)=q(t)I+\alpha(t)E,\qquad D(t)=\delta(t)I \quad \text{for } t_0 \leqslant t \leqslant T, \]

where the scalar functions \(q(t), \alpha(t), \delta(t)\) are continuous on \([t_0,T]\), then

\[ \left| \Omega_{t_0}^{t} B-\Omega_{t_0}^{t} A \right| \leqslant \exp\int_{t_0}^{t}\alpha\,du \cdot \exp\left(\int_{t_0}^{t} q\,du\, I\right) \cdot \left\{ \exp\left(\int_{t_0}^{t}\delta\,du\, I\right)-E \right\} \quad \text{for } t_0 \leqslant t \leqslant T . \]

1. Theorem 2 (estimate of the norm of the difference of multiplicative integrals). Suppose there exist real functions \(g(t), M(t), L(t), \delta(t)\), continuous on \([t_0,T]\) and satisfying the conditions

\[ \left\| \Omega_{\xi}^{t} A \right\| \leqslant M(t)L(\xi)\exp\int_{\xi}^{t} g(u)\,du \quad \text{for } t_0 \leqslant \xi \leqslant t \leqslant T, \tag{3} \]

\[ \|B(t)-A(t)\|\leqslant \delta(t) \quad \text{for } t_0 \leqslant t \leqslant T . \tag{4} \]

Then

\[ \left\| \Omega_{t_0}^{t} B-\Omega_{t_0}^{t} A \right\| \leqslant M(t)L(t_0)\exp\int_{t_0}^{t} g\,du \cdot \left\{ \exp\int_{t_0}^{t} ML\delta\,du-1 \right\} \quad \text{for } t_0 \leqslant t \leqslant T . \tag{5} \]

Proofs.

1. Suppose \(|A_k|\leqslant P_k,\ |B_k-A_k|\leqslant D_k\) for \(k=1,\ldots,N\). Then

\[ |B_NB_{N-1}\cdots B_1-A_NA_{N-1}\cdots A_1| \leqslant \]

\[ \leqslant (P_N+D_N)(P_{N-1}+D_{N-1})\cdots(P_1+D_1)-P_NP_{N-1}\cdots P_1 . \]

Proof. For \(N=1\) this is obvious. Noting that

\[ B_{N+1}B_N\cdots B_1-A_{N+1}A_N\cdots A_1= \]

\[ = (B_{N+1}-A_{N+1})B_N\cdots B_1 + A_{N+1}(B_N\cdots B_1-A_N\cdots A_1), \]

we apply induction on \(N\).

1. Suppose \(R\) is real, \(S\) is a real diagonal matrix, \(\Delta\) is a nonnegative number, and

\[ [R]_{\mu\mu}\geqslant -\frac12,\qquad |[S]_{\mu\mu}|\leqslant \Delta \quad \text{for } \mu=1,\ldots,n; \]

\[ [R]_{\mu\nu}\geqslant 0 \quad \text{for } \mu,\nu=1,\ldots,n;\ \mu\neq\nu . \]

Then

\[ |E+R+iS|\leqslant (1+2\Delta^2)(E+R). \]

Proof. By elementary calculations.

1. Proof of Theorem 1. Define the diagonal matrix-function \(S\) by the formulas

\[ [S(t)]_{\mu\mu}\overset{\mathrm{def}}{=}\operatorname{Im}[A(t)]_{\mu\mu}, \quad \text{for } \mu=1,\ldots,n. \]

Find a nonnegative number \(\Delta\) and a natural number \(N_0\) such that

\[ |[S(t)]_{\mu\mu}|\leqslant \Delta,\qquad \frac{T-t_0}{N_0}[A^{+}(t)]_{\mu\mu}\geqslant -\frac12 \quad \text{for } \mu=1,\ldots,n;\ t_0\leqslant t\leqslant T . \tag{6} \]

Fix \(t,\ t_0\leqslant t\leqslant T\), and a natural number \(N>N_0\). Put

\[ h\overset{\mathrm{def}}{=}\frac{t-t_0}{N},\qquad t_k\overset{\mathrm{def}}{=}t_0+kh \quad \text{for } k=1,\ldots,N . \tag{7} \]

Finally, set \(A_k\overset{\mathrm{def}}{=}A(t_k)\) for \(k=1,\ldots,N\), and analogously for

matrix-functions \(B, C, D, S\). By item 6, (6), and (7),

\[ \begin{gathered} \left|E+hA_k\right|=\left|E+hA_k^{+}+ihS_k\right|\leq\\ \leq (1+2h^2\Delta^2)(E+hA_k^{+})\leq (1+2h^2\Delta^2)(E+hC_k)\\ \text{for } k=1,\ldots,N . \end{gathered} \tag{8} \]

By (8) and the second of inequalities (2),

\[ \begin{gathered} \left|E+hB_k\right|\leq \left|E+hA_k\right|+h\left|B_k-A_k\right| \leq \left|E+hA_k\right|+hD_k\leq\\ \leq (1+2h^2\Delta^2)(E+h(C_k+D_k)) \quad \text{for } k=1,\ldots,N . \end{gathered} \]

Replacing, in item 5, \(A_k\) by \(E+hA_k\), \(B_k\) by \(E+hB_k\), \(P_k\) by \((1+2h^2\Delta^2)\times (E+hC_k)\), and \(D_k\) by \((1+2h^2\Delta^2)hD_k\), we obtain

\[ \begin{gathered} \left|(E+hB_N)\ldots(E+hB_1)-(E+hA_N)\ldots(E+hA_1)\right|\leq\\ \leq (1+2h^2\Delta^2)^N\{(E+h(C_N+D_N))\ldots(E+h(C_1+D_1))-\\ -(E+hC_N)\ldots(E+hC_1)\}. \end{gathered} \]

Letting \(N\to\infty\), we obtain a); b) follows from a) \(((2), p. 386)\); c) follows from b).

8.

\[ \left\|\Omega_{\xi}^{t} A\right\|\leq \exp\int_{\xi}^{t}\|A(u)\|\,du \quad \text{for } t_0\leq \xi\leq t\leq T . \]

Proof. This follows from the representation of the matriciant as the limit of an integral product \(((2), Ch. XIV, formula (49'))\).

1. Let \(A\) and \(B\) be matrix-functions on \([t_0,T]\). Define the matrix-function \(D\) by the formula

\[ D(t)\overset{\mathrm{def}}{=}B(t)-A(t) \quad \text{for } t_0\leq t\leq T . \tag{9} \]

Then

\[ \Omega_{t_0}^{t} B = \Omega_{t_0}^{t} A + \int_{t_0}^{t} \left(\Omega_{\xi}^{t} A\right)D(\xi) \left(\Omega_{t_0}^{\xi} B\right)\,d\xi \quad \text{for } t_0\leq t\leq T . \tag{10} \]

Proof. Let \(L(t)\) denote the left-hand side, and \(\Pi(t)\) the right-hand side of formula (10). Then a simple calculation shows that the matrix-functions \(L\) and \(\Pi\) are solutions of the Cauchy problem

\[ \frac{dX}{dt}=A(t)X+D(t)\Omega_{t_0}^{t}B,\quad X(t_0)=E \tag{11} \]

(\(X\) is the unknown matrix-function, \(t_0\leq t\leq T\)). By uniqueness of the solution of problem (11), we have \(L(t)=\Pi(t)\) for \(t_0\leq t\leq T\), as was required to prove.

1. Proof of Theorem 2. Keeping (9), define the matrix-functions \(I_k, I_k^*\) \((k=0,1,2,\ldots)\) by the formulas

\[ I_0(t)\overset{\mathrm{def}}{=}\Omega_{t_0}^{t} A, \quad I_0^*(t)\overset{\mathrm{def}}{=}\Omega_{t_0}^{t} B, \tag{12} \]

\[ I_{k+1}\overset{\mathrm{def}}{=} \int_{t_0}^{t} \left(\Omega_{\xi}^{t} A\right)D(\xi)I_k(\xi)\,d\xi \quad \text{for } k=0,1,2,\ldots, \tag{13} \]

\[ I_{k+1}^*\overset{\mathrm{def}}{=} \int_{t_0}^{t} \left(\Omega_{\xi}^{t} A\right)D(\xi)I_k^*(\xi)\,d\xi \quad \text{for } k=0,1,2,\ldots, \tag{14} \]

where \(t_0\leq t\leq T\). Then

\[ \Omega_{t_0}^{t} B = I_0(t)+\ldots+I_k(t)+I_{k+1}^*(t) \quad \text{for } k=0,1,2,\ldots;\ t_0\leq t\leq T . \]

Indeed, for \(k=0\) this follows from (12), (14), and item 9; next we apply (using (10)) induction on \(k\).

Denote by \(M\) some positive number such that

\[ \left\|\Omega_{t_0}^{t} B\right\|\le M \quad \text{for } t_0\le t\le T . \tag{15} \]

Then

\[ \left\|I_{k+1}^{*}(t)\right\|\le M\exp\int_{t_0}^{t}\|A(u)\|\,du\cdot \frac{1}{(k+1)!}\left\{\int_{t_0}^{t}\delta(u)\,du\right\}^{k+1} \tag{16} \]

for \(k=0,1,2,\ldots;\ t_0\le t\le T\).

Indeed, for \(k=0\) inequality (16) follows from item 8, (15), (14), and (12). Next we apply induction on \(k\). From (16) it follows that \(\|I_{k+1}^{*}(t)\|\to 0\) as \(k\to\infty,\ t_0\le t\le T\), and consequently,

\[ \Omega_{t_0}^{t}B=\sum_{k=0}^{\infty} I_k(t) \quad \text{for } k=0,1,2,\ldots . \tag{17} \]

Let us now note that

\[ \left\|I_k(t)\right\|\le M(t)L(t_0)\exp\int_{t_0}^{t}g(u)\,du\cdot \frac{1}{k!}\left\{\int_{t_0}^{t}ML\delta\,du\right\}^{k} \tag{18} \]

for \(k=0,1,2,\ldots;\ t_0\le t\le T\).

Indeed, for \(k=0\) this is true by virtue of (12) and (3); we apply (using (13), (4), and (9)) induction on \(k\). From (17) it follows that

\[ \left\|\Omega_{t_0}^{t}B-\Omega_{t_0}^{t}A\right\| \le \sum_{k=1}^{\infty}\left\|I_k(t)\right\| \quad \text{for } t_0\le t\le T . \tag{19} \]

From (18) and (19) we obtain (5).

Additional Remarks

11. Let \(A\) and \(C\) be matrix-functions on \([t_0,T]\), with \(A^{+}(t)\le C(t)\) for \(t_0\le t\le T\). Then

\[ \left|\Omega_{t_0}^{t} A\right|\le \Omega_{t_0}^{t} C \quad \text{for } t_0\le t\le T . \]

Proof. With the notation of item 7, (8) is true, which gives

\[ \left|(E+hA_N)\cdots(E+hA_1)\right| \le (1+2h^2\Delta^2)^N(E+hC_N)\cdots(E+hC_1). \]

Let \(N\to\infty\).

12. Theorem 1b) and item 11 generalize known results \(\left((^{2}),\ \text{Ch. XIV},\ §6,\ \text{VIII and } §5,\ 5^\circ;\ \text{there—in different notation—} |A(t)|\le C(t),\ \alpha(t)=0,\ q(t)\ \text{and } \delta(t)\ \text{constant}\right)\). Condition (3) in Theorem 2 is fulfilled, for example, if \(M(t)=L(t)=1,\ g(t)=\gamma(A(t))\), where \(\gamma\) is the logarithmic norm of the matrix corresponding to the introduced matrix norm \(\left((^{3}),\ \text{pp. }57\text{--}62\right)\).

13. The linear transformation \(x=U(t)y\), where the matrix-function \(U(t)\) has a continuous derivative and is nonsingular for \(t_0\le t\le T\), transforms system (1) into a system with matrix \(Q(t)\), and

\[ \Omega_{\xi}^{t}A = U(t)\left(\Omega_{\xi}^{t}Q\right)U^{-1}(\xi). \]

Hence it is easy to obtain a more general case of the fulfillment of (3) than in item 12.

14. If the matricant is defined as the sum of a series \(\left((^{2}),\ \text{Ch. XIV},\ \text{formula }(10)\right)\), then in Theorems 1 and 2 it is sufficient, instead of continuity of the matrix-functions, to assume summability.

Received
17 IV 1966

References

\({}^{1}\) D. K. Faddeev, V. N. Faddeeva, Computational Methods of Linear Algebra, 1960.
\({}^{2}\) F. R. Gantmacher, The Theory of Matrices, 1953.
\({}^{3}\) S. M. Lozinskii, Izv. Vyssh. Uchebn. Zaved., Matematika, No. 5, 52 (1958).