Lyapunov Exponents for Matrices with Invariant Subspaces

Abstract

If $\mathbf{M}$ is a fixed $d \times d$ complex-valued matrix, then the eigenvalues of $\mathbf{M}'$, the conjugate transpose of $\mathbf{M}$, are the complex conjugates of the eigenvalues of $\mathbf{M}$, with the same multiplicities, and if $\mathbf{M}$ is upper block triangular, the eigenvalues of $\mathbf{M}$ are the eigenvalues of the diagonal blocks, and the multiplicities add. We shall show that if $\{\mathbf{M}(j)\}$ is a stationary, ergodic, time-reversible sequence taking values in the $d \times d$ complex matrices, then similar properties hold for the Lyapunov exponents of $\{\mathbf{M}(j)\}$.