Let 𝐮=𝐱+i 𝐲 be a complex solution to a real linear system....

Let $\mathbf{u}=\mathbf{x}+\mathrm{i} \mathbf{y}$ be a complex solution to a real linear system. Under what conditions are its real and imaginary parts $\mathbf{x}, \mathbf{y}$ linearly independent real solutions?

Key Concepts

Complex Eigenvalues in Real Systems

In real linear systems, complex eigenvalues occur in conjugate pairs. This means if a matrix has a complex eigenvalue, its corresponding eigenvector is also complex. The presence of these complex conjugate pairs is essential because they allow the construction of real solutions by separating the complex eigenvector into its real and imaginary parts.

Real and Imaginary Parts of Complex Solutions

A complex solution to a real system can be written as the sum of its real part and an imaginary part multiplied by the imaginary unit. Each of these parts, when taken separately, still satisfies the system due to the system’s linearity and real coefficients. Thus, they provide valid real solutions on their own.

Linear Independence of Real and Imaginary Solutions

For the real and imaginary parts of a complex solution to form a fundamental set of solutions, they must be linearly independent. This means that one cannot be obtained from the other by a simple scalar multiplication. Their linear independence is crucial for spanning the solution space, particularly when dealing with systems having non-real eigenvalues.

Condition of Non-Real Eigenvalues

The requirement that the eigenvalue associated with the complex solution is non-real (i.e., it has a nonzero imaginary part) is necessary for the linear independence of the real and imaginary parts. If the eigenvalue were real, the solution would be entirely real and its imaginary part would vanish, failing to provide a second independent solution.

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