  This article introduces a nonparametric approach to multivariate time-varying
power spectrum analysis. The procedure adaptively partitions a time series into
an unknown number of approximately stationary segments, where some spectral
components may remain unchanged across segments, allowing components to evolve
differently over time. Local spectra within segments are fit through Whittle
likelihood based penalized spline models of modified Cholesky components, which
provide flexible nonparametric estimates that preserve positive definite
structures of spectral matrices. The approach is formulated in a Bayesian
framework, in which the number and location of partitions are random, and
relies on reversible jump Markov chain and Hamiltonian Monte Carlo methods that
can adapt to the unknown number of segments and parameters. By averaging over
the distribution of partitions, the approach can approximate both abrupt and
slow-varying changes in spectral matrices. Empirical performance is evaluated
in simulation studies and illustrated through analyses of
electroencephalography during sleep and of the El Ni\~no-Southern Oscillation.
