Actually, one can easily restore from the matrix-valued function by a simple limiting process.

For instance, the Fourier transform yields a scalar- and not matrix-valued function.

So by extending the definition of PR to matrix-valued functions, linear multi-port networks which are passive may be distinguished from those that are not.

One can show that a matrix-valued function is a fundamental matrix of if and only if and is a non-singular matrix for all .

This enabled him to show that the algebra of bounded singular integral operators is isomorphic to the algebra of either scalar or matrix-valued functions.

The same holds if "f" and "g" are matrix-valued functions (so that the corresponding Toeplitz operators are matrices of operators on H("S")).

The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra, where the last equality is no longer true (see the references below).

Application of the theory: functions defined as pointwise maxima of a family of functions; minimizing the maximal eigenvalue of a matrix-valued function; variational analysis of an extended eigenvalue problem.

Suppose that F(z) is a complex analytic matrix-valued function on a rectangle K in C such that F(z) is an invertible matrix for each z in K.

Here a commonly stated definition of effective mass is the inertial effective mass tensor defined below; however, in general it is a matrix-valued function of the wavevector, and even more complex than the band structure itself.