Common Conjugation Errors In Eigenvalue Computations

Conjugate the Whole Eigenvalue Equation

Complex eigenvalue calculations often involve a short step that is easy to mishandle: taking the complex conjugate of an equation. Conjugation applies to every scalar and every matrix entry in that equation. In particular, it changes \(\lambda\) to \(\overline{\lambda}\), and it changes a complex matrix \(\mathbf{A}\) to its entrywise conjugate \(\overline{\mathbf{A}}\). It does not, by itself, transpose a matrix.

Throughout this tutorial, vectors are columns in \(\mathbb{C}^n\). A complex number \(a+bi\), with \(a,b\in\mathbb{R}\), has conjugate \(a-bi\). For a vector or matrix, the overline means that conjugation is applied entry by entry. An eigenvalue and eigenvector of \(\mathbf{A}\in\mathbb{C}^{n\times n}\) satisfy \(\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\), where \(\mathbf{v}\ne\mathbf{0}\).

Proof. Write the \(j\)th entry of \(\mathbf{A}\mathbf{v}\) as \(\sum_{k=1}^{n}a_{jk}v_k\). Conjugating the \(j\)th component of the given equation gives \(\overline{\sum_{k=1}^{n}a_{jk}v_k}=\overline{\lambda v_j}\). Conjugation of a sum and a product gives \(\sum_{k=1}^{n}\overline{a_{jk}}\,\overline{v_k}=\overline{\lambda}\,\overline{v_j}\). This holds for every row \(j\), so it is exactly \(\overline{\mathbf{A}}\,\overline{\mathbf{v}}=\overline{\lambda}\,\overline{\mathbf{v}}\). Since \(\mathbf{v}\ne\mathbf{0}\), at least one entry of \(\mathbf{v}\) is nonzero, and its conjugate is nonzero; hence \(\overline{\mathbf{v}}\ne\mathbf{0}\). If \(\mathbf{A}\) is real, then \(\overline{\mathbf{A}}=\mathbf{A}\), giving the stated eigenpair of the same matrix. \(\square\)

The hypothesis that \(\mathbf{A}\) is real matters. For a complex matrix, conjugating an eigenpair generally gives an eigenpair of \(\overline{\mathbf{A}}\), not necessarily of \(\mathbf{A}\). For a real matrix, by contrast, a nonreal eigenvalue \(\lambda\) is accompanied by \(\overline{\lambda}\). This conclusion concerns eigenvalues and their corresponding eigenvectors together; it does not say that \(\overline{\lambda}\) can be paired with the original vector \(\mathbf{v}\).

Worked Examples of Correct Conjugation

Transpose Is Not Conjugation

For a complex matrix, the transpose \(\mathbf{A}^{\top}\) swaps rows and columns without changing entries. The conjugate transpose \(\mathbf{A}^{*}=\overline{\mathbf{A}}^{\top}\) first conjugates entries and also swaps rows and columns. These operations are distinct. In particular, the step from \(\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\) to \(\overline{\mathbf{A}}\,\overline{\mathbf{v}}=\overline{\lambda}\,\overline{\mathbf{v}}\) uses entrywise conjugation, not a transpose.

For a real matrix, \(\overline{\mathbf{A}}=\mathbf{A}\), and therefore \(\mathbf{A}^{*}=\mathbf{A}^{\top}\). Even in that case, taking the transpose is not the operation used to conjugate an eigenvalue equation. The operations may happen to coincide in one comparison, but their definitions and roles remain different.

Complex Eigenvectors Are Not Automatically Real Eigenvectors

A vector in \(\mathbb{C}^n\) can be written uniquely as \(\mathbf{x}=\mathbf{u}+i\mathbf{w}\), where \(\mathbf{u},\mathbf{w}\in\mathbb{R}^n\) are its real and imaginary parts. If a real matrix acts on this vector, it acts on both parts. That does not mean either part separately is an eigenvector for the original complex eigenvalue.

Proof. Since \(\mathbf{A}\) is real, multiplication distributes over the real and imaginary parts: \(\mathbf{A}\mathbf{x}=\mathbf{A}\mathbf{u}+i\mathbf{A}\mathbf{w}\). Also, \((a+bi)(\mathbf{u}+i\mathbf{w})=(a\mathbf{u}-b\mathbf{w})+i(b\mathbf{u}+a\mathbf{w})\), using \(i^2=-1\). Two complex vectors are equal exactly when their real parts are equal and their imaginary parts are equal. Thus \(\mathbf{A}\mathbf{x}=\lambda\mathbf{x}\) holds exactly when both displayed real-vector equations hold. \(\square\)

There is also a direct reason a real vector cannot be an eigenvector of a real matrix for a nonreal eigenvalue. If \(\mathbf{r}\in\mathbb{R}^n\) is nonzero and \(\mathbf{A}\mathbf{r}=\lambda\mathbf{r}\), then both \(\mathbf{A}\mathbf{r}\) and \(\mathbf{r}\) are real. Choose an index \(j\) with \(r_j\ne0\). The \(j\)th coordinate gives \(\lambda=(\mathbf{A}\mathbf{r})_j/r_j\), a real number. Hence \(\lambda\) must be real. This argument requires \(\mathbf{A}\) and \(\mathbf{r}\) to be real.

Common Conjugation Errors

Error: change the eigenvalue but keep the eigenvector. Starting from \(\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\), one might write \(\mathbf{A}\mathbf{v}=\overline{\lambda}\mathbf{v}\) for a real matrix. In the rotation example, this would claim \(\mathbf{A}\begin{bmatrix}1\\-i\end{bmatrix}=-i\begin{bmatrix}1\\-i\end{bmatrix}\). The left side is \(\begin{bmatrix}i\\1\end{bmatrix}\), while the right side is \(\begin{bmatrix}-i\\-1\end{bmatrix}\); they are not equal. The correction is to conjugate the vector as well: the valid equation is \(\mathbf{A}\begin{bmatrix}1\\i\end{bmatrix}=-i\begin{bmatrix}1\\i\end{bmatrix}\).

Error: use the real and imaginary parts as separate eigenvectors. For \(\mathbf{x}=\mathbf{u}+i\mathbf{w}\), the eigenvector equation is equivalent to the coupled equations in the theorem, not to \(\mathbf{A}\mathbf{u}=\lambda\mathbf{u}\) and \(\mathbf{A}\mathbf{w}=\lambda\mathbf{w}\). In particular, those latter equations would equate real vectors with generally nonreal vectors when \(\lambda\) is nonreal.

Error: replace conjugation by transpose. A transpose swaps matrix entries across the diagonal; it does not conjugate \(i\) to \(-i\). When conjugating an eigenvalue equation, conjugate every entry and scalar, and do not transpose unless a separate argument calls for a transpose or conjugate transpose.

A reliable check is to return to the defining equation and multiply the proposed matrix by the proposed vector. For a conjugate pair of a real matrix, check both \(\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\) and \(\mathbf{A}\overline{\mathbf{v}}=\overline{\lambda}\,\overline{\mathbf{v}}\). For a complex matrix, record whether the equation involves \(\mathbf{A}\), \(\overline{\mathbf{A}}\), \(\mathbf{A}^{\top}\), or \(\mathbf{A}^{*}\); these are not interchangeable.