Complex matrices are those that contain complex numbers, while real matrices only contain real numbers. Complex numbers have both a real and imaginary component, while real numbers only have a real component.
A matrix is complex if it contains at least one complex number in any of its entries. If all entries are real numbers, then the matrix is considered real.
No, a real matrix can only have real eigenvalues. This is because the eigenvalues of a matrix are the roots of its characteristic polynomial, and the coefficients of a real matrix's characteristic polynomial can only be real numbers.
Complex matrices are commonly used in engineering, physics, and other scientific fields to model systems that involve both real and imaginary components. They are also used in signal processing and control systems.
To solve a system of equations with complex matrices, the same methods used for real matrices can be applied. However, it is important to be aware of the properties of complex numbers, such as the fact that complex conjugates must be taken when dividing by a complex number.