Semiimaginary

Semiimaginary refers to a property of certain mathematical objects, particularly matrices and numbers, indicating that their real part is zero and their imaginary part is non-zero. This concept is most commonly encountered in the context of complex numbers. A complex number of the form $bi$, where $b$ is a non-zero real number, is considered semiimaginary. For example, $3i$ and $-2i$ are semiimaginary numbers.

The term can also be extended to describe matrices. A matrix is sometimes called semiimaginary if all its entries are semiimaginary numbers. Alternatively, in some contexts, a matrix might be considered semiimaginary if its real part is the zero matrix and its imaginary part is a non-zero matrix. This usage is less standardized than its application to complex numbers.

In the study of differential equations and dynamical systems, the eigenvalues of a system can be semiimaginary. When an eigenvalue is purely imaginary (i.e., of the form $i\omega$ where $\omega$ is a non-zero real number), it often indicates oscillatory behavior in the system. This can lead to phenomena such as limit cycles or stable oscillations. The term "semiimaginary" in this context emphasizes the absence of a real component that would cause growth or decay, focusing solely on the oscillatory nature.