  In this paper, we define the phases of a complex sectorial matrix to be its
canonical angles, which are uniquely determined from a sectorial decomposition
of the matrix. Various properties of matrix phases are studied, including those
of compressions, Schur complements, matrix products, and sums. In particular,
by exploiting a notion known as the compound numerical range, we establish a
majorization relation between the phases of the eigenvalues of $AB$ and the
phases of $A$ and $B$. This is then applied to investigate the rank of $I + AB$
with phase information of $A$ and $B$, which plays a crucial role in feedback
stability analysis. A pair of problems: banded sectorial matrix completion and
decomposition is studied. The phases of the Kronecker and Hadamard products are
also discussed.
