42 Infinite product of matrices
If \(\{B_n\}_{n \ge 1}\) are \(p \times p\) real matrices, we define \[ \prod_{k=n}^m B_k = \begin{cases} B_m B_{m-1} \cdots B_n & \hbox{if } n \le m, \\ I & \hbox{if } n > m; \end{cases} \] to be the product where successive terms multiply on the left.
We say that an infinite product \(\prod_{n=1}^\infty B_n\) of \(p \times p\) matrices converges if there exists an integer \(N\) such that \[ Q = \lim_{m \to \infty} \prod_{n=N}^m B_n \] exists. In this case, we define \(\prod_{n=1}^\infty B_n = Q \prod_{n=1}^{N-1} B_n\).
Following Trench (1999), we say that the product \(\prod_{n=1}^\infty B_n\) converges invertibly if for all \(n \ge N\), \(B_n\) is invertible and the product \(Q\) defined above is invertible as well. It was argued in Trench (1999) that the above definition has the following consequences:
- (P1) An invertibly convergent infinite product is singular if and only if at least one of its factors is singular.
- (P2) If \(\prod_{n=1}^\infty B_n\) converges invertibly then \(\lim_{n \to \infty} B_n = I\).
We now present some sufficient conditions for convergence of infinite product of matrices. If the matrices being multiplied are invertible, then the product converges to an invertible limit. Let \(\NORM{\cdot}\) denote any sub-multiplicative matrix norm on \(\reals^{p \times p}\)
(C1) If \[ \sum_{n=1}^{\infty} \NORM{B_n} < \infty \] then \(\prod_{n=1}^\infty (I + B_n)\) converges.
(C2) Let \(\{R_n\}_{n \ge 1}\) be a sequence of \(p \times p\) matrices such that \[ \lim_{n \to \infty} R_n = I \] and \[ \sum_{n=1}^{\infty} \NORM{ (I + B_n) R_n - R_{n+1} } < \infty \] then \(\prod_{n=1}^{\infty}(I + B_n)\) converges.
(C3) Let \(\{U_n\}_{n \ge 1}\) be a sequence of \(p \times p\) matrices such that \(\NORM{U_n} = 1\) for all \(n\), there exists a \(N\) such that \(\prod_{n=N}^m U_n\) converges for \(m \to \infty\), and \[ \sum_{n=1}^{\infty} \NORM{B_n} < \infty \] then \(\prod_{n=1}^\infty (U_n + B_n)\) converges.
Exercises
Exercise 42.1 Suppose \(A_t = \MATRIX{0 & 1 + \beta_t \\ 1 & \gamma_t}\), where \(\beta_t, \gamma_t \in [0,1]\) Show that \(\prod_{t \ge 1} A_t\) converges if \[ \sum_{t \ge 1} (\beta_t + \gamma_t) < \infty. \]
Hint: Write \(A_t = P + B_t\), where \(P = \MATRIX{0 & 1 \\ 1 & 0}\) is a permultation matrix. So \(P^2 = I\). Break the product \(\prod_{t \ge 1} A_t\) as \[ \cdots (A_4 A_3) (A_2 A_1) \] and write \(A_{2t}A_{2t-1}\) in the form \(I + C_t\) and use property (C1).
Notes
Condition (C1) is a standard result and stated as Theorem 1 in Trench (1999). Condition (C2) is Theorem 5 of Trench (1999). Condition (C3) is Theorem 2.1 combined with the remark on page 15 of Artzrouni (1986).
Exercise 42.1 is from Mahajan et al. (2024).