Yu. S. Bogdanov

LYAPUNOV NORMS IN LINEAR SPACES

(Presented by Academician V. I. Smirnov on 6 X 1956)

MATHEMATICS

1°. Notation. Let us denote: \(A\) — a linear set, \(a\) — elements of \(A\); \(T\) — an automorphism of \(A\); \(B\) — an \(n\)-dimensional subspace of \(A\); \(b\) — elements of \(B\); \(\beta \equiv (b^1, b^2, \ldots, b^n)\) — a basis of \(B\); \(\Delta\) — an ordered set; \(\lambda\) — a mapping of \(A\) into \(\Delta\); \(c\) — a real number; \(C \equiv (c_j^i)\) — an \(n\)-dimensional matrix \((i\) — column number, \(j\) — row number\()\).

2°. Definition of a Lyapunov norm. We shall say that \(\lambda\) normizes \(A\) in the sense of Lyapunov if, for any \(a, a', c\):

1) \(\lambda(ca) \preceq \lambda(a)\);

2) \(\lambda(a+a') \preceq \max\{\lambda a,\lambda a'\}\).

The element \(\lambda a \in \Delta\) will be called the Lyapunov norm \(\lambda\) (or the \(\lambda\)-norm) of the element \(a \in A\).

3°. Properties of the Lyapunov norm. From the definition of the Lyapunov norm it follows:

a) \(c \ne 0 \to \lambda(ca)=\lambda a\);

b) \(\lambda a \succeq \lambda0\);

c) \(\lambda(c_1a^1+c_2a^2+\cdots+c_ma^m)=\preceq \max(\lambda a^1,\lambda a^2,\ldots,\lambda a^m)\);

d) \(c \ne 0,\ \lambda a \succ \lambda a^i\ (i=1,2,\ldots,m)\to \lambda(ca+c_1a^1+\cdots+c_ma^m)=\lambda a\);

e) \(\lambda a^i \ne \lambda a^j,\ i\ne j,\ a^i\ne0\ (i,j=1,2,\ldots,m)\to a^1,a^2,\ldots,a^m\) are linearly independent.

4°. The Lyapunov norm in a finite-dimensional linear space. Any \(m\) \((m>n)\) elements of \(B\) are linearly dependent, and therefore (see 3°, e)) the set \(\{\lambda b\}\), where \(b\) is any nonzero element of \(B\), has no more than \(n\) pairwise distinct elements.

5°. Definition of a \(\lambda\)-basis. A basis \(\beta\) of the set \(B\) will be called a \(\lambda\)-basis if every linear combination of the elements of \(\beta\) has a \(\lambda\)-norm equal to the greatest of the \(\lambda\)-norms of those elements of \(\beta\) which enter into the combination with coefficients different from 0.

6°. Existence of a \(\lambda\)-basis. Let \(\beta\) be a basis of \(B\). Then there exists a triangular matrix \(C\) with unit diagonal elements such that \(\beta_1=\beta_2C\) is a \(\lambda\)-basis of \(B\). The proof of the statement just made is carried out in the same way as the proof of Lyapunov’s theorem on the existence of a triangular matrix \(C\) transforming a given fundamental system of solutions of a system of homogeneous linear differential equations into a normal system of solutions \((^1)\).

7°. Properties of a \(\lambda\)-basis. Here and in the remaining paragraphs let \(\beta_1\) denote a \(\lambda\)-basis of \(B\); \(\beta_2\) a basis of \(B\), with any basis assumed to be written so that \(\lambda b_i^j \preceq \lambda b_i^l\), if \(j<l\) \((i,j=1,2,\ldots,n)\). Then:

a) \(0\ne b\in B\to \lambda b\in\{\lambda b_1^1,\lambda b_1^2,\ldots,\lambda b_1^n\}\);

b) \(\lambda b\preceq \lambda b_1^i\to b=c_1b_1^1+c_2b_1^2+\cdots+c_{i-1}b_1^{i-1};\)

c) in order that the basis \(\beta_2\) be a \(\lambda\)-basis of \(B\), it is necessary and sufficient that
\(\lambda b_2^i=\lambda b_1^i,\ i=1,2,\ldots,n\).

8°. The number of elements of a basis with one and the same \(\lambda\)-norm. Let \(k\) denote the number of mutually distinct values \(\lambda b_1^i\) \((i=1,2,\ldots,n)\). Suppose that these values are \(\delta_1,\delta_2,\ldots,\delta_k\), with \(\delta_i\succ \delta_j\) for \(i<j\). Suppose further that \(n_i\) is the greatest possible number of linearly independent elements of \(B\) with Lyapunov norm not exceeding \(\delta_i\). Denote by \(N_i(\beta_2)\) the number of elements of \(\beta_2\) with \(\lambda\)-norm \(\delta_i\). Then:

a) \(N_1(\beta_2)+N_2(\beta_2)+\cdots+N_k(\beta_2)=n\);

b) \(n_k=n,\ n_i<n_j\), if \(i<j\);

c) \(N_1(\beta_2)+N_2(\beta_2)+\cdots+N_i(\beta_2)\leq n_i\);

d) \(N_1(\beta_1)=n_1,\ N_i(\beta_1)=n_i-n_{i-1}\ (i=2,3,\ldots,k)\).

9°. Step matrices. A matrix \(C\equiv(c_j^i)\) will be called \(m_l;\,r\)-step if \(c_j^i=0\) for \(i\leq m_l<j\) for any \(l=1,2,\ldots,r;\ i,j=1,2,\ldots,n\) \((m_r=n)\). A triangular matrix is, evidently, \(l;\,n\)-step. All nonsingular \(m_l;\,r\)-step matrices (for fixed \(m_l\)) form a group with respect to multiplication in the usual sense for matrices.

10°. Transformation of a \(\lambda\)-basis. The basis \(\beta_2\), like any other system of \(n\) elements of \(B\), can be represented in the form \(\beta_2=\beta_1 C\), where \(C\) is a real matrix. The basis \(\beta_2\) will be a \(\lambda\)-basis if and only if \(C\) is an \(n_i;\,k\)-step nonsingular matrix. This can be proved by the same arguments by which the validity of the corresponding proposition is established for a normal system of solutions \((^2)\).

11°. The sets \(M_i\). Denote by \(M_i\) the set of all elements of \(B\) with \(\lambda\)-norm \(\delta_i\), where \(0\) is not included in \(M_1\), even if \(\lambda 0=\delta_1\). In addition, put \(M_0\equiv0\). Then:

a) \(M_i\cap M_j\) for \(i\ne j\);

b) \(M_0\cup M_1\cup\cdots\cup M_k=B\).

12°. The structure of \(M_i\). Each \(M_i\) is an \(n_i\)-dimensional hyperplane with the \(n_{i-1}\)-dimensional hyperplane removed. Indeed,
\[ M_0\cup M_1\cup\cdots\cup M_i=\bigcup \left(c_1 b_1^1+c_2 b_1^2+\cdots+c_{n_i}b_1^{n_i}\right) \]
(the last sum is taken over all \(c_1,c_2,\ldots,c_{n_i}\in(-\infty,\infty)\)). The converse assertion is also true: if \(B\) is represented as the sum of hyperplanes \(P_0\equiv0;\ P_1,P_2,\ldots,P_m\), each of which strictly contains the preceding ones, then one can indicate a Lyapunov norm \(\lambda\) such that the decomposition of the space \(B\) into the sets \(M_1,M_2,\ldots,M_k\) effected by it has the property: \(k=m;\ M_i=P_i-P_{i-1}\ (i=1,2,\ldots,k)\).

13°. \(\lambda\)-similar transformations of \(A\). An automorphism \(T\) of the linear set \(A\): \(TA=A\), will be called a \(\lambda\)-similar transformation if
\[ \lambda a \preceq \max\{\lambda Ta,\lambda T^{-1}a\}. \]
In order that \(T\) be a \(\lambda\)-similar transformation, it is necessary and sufficient that
\[ \lambda a=\lambda Ta=\lambda T^{-1}a \]
(\(a\), as always, is any element of \(A\)).

14°. Invariants of \(\lambda\)-similar transformations. Under an automorphism \(T\) of the linear set \(A\), an \(n\)-dimensional linear subset \(B\subset A\) is transformed into an \(n\)-dimensional linear subset \(\widetilde B\subset A\). It is not difficult to show (based, for example, on 7°, c)) that if \(T\) is a \(\lambda\)-similar transformation, then the \(\lambda\)-basis \(\beta_1\) of the linear subset \(B\) under \(T\) passes into the \(\lambda\)-basis \(\widetilde\beta_1\) of the linear subset \(\widetilde B\), and moreover
\(\lambda_1 b_1^i=\lambda\widetilde b_1^i\ (i=1,2,\ldots,n)\). Thus the numbers \(n_i\) (see 8°) are invariants of all \(\lambda\)-similar transformations.

15°. Examples. The first example of a Lyapunov norm on the set of solutions of all possible systems of homogeneous linear differential equations specified for \(t \geqslant t_0\) is the Lyapunov characteristic number \((^3)\), taken with the opposite sign. As a second example one may point to the aggregate (characteristic exponent, type of solution) considered by L. Markus \((^4)\). In this case the second \([\lambda'']\)-norm refines the first \(\{\lambda'\}\), i.e.,
\[ \lambda'' a \supseteq \lambda'' a' \to \lambda' a \supseteq \lambda' a', \qquad \lambda'' a \subseteq \lambda'' a' \to \lambda' a = \lambda' a' . \]

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
3 X 1956

REFERENCES

\(^{1}\) A. M. Lyapunov, The General Problem of the Stability of Motion, Moscow—Leningrad, 1950, pp. 48–50.
\(^{2}\) Yu. S. Bogdanov, DAN, 57, No. 3, 215 (1947).
\(^{3}\) A. M. Lyapunov, DAN, 57, No. 3, 39 (1947).
\(^{4}\) L. Markus, Math. Zs., 62, 310 (1955).