Worked Problems On Real 2x2 Matrices With Complex Eigenvalues

From a Complex Eigenpair to a Real Matrix Form

A real matrix can have nonreal eigenvalues, even though it sends real vectors to real vectors. In this tutorial, the complex eigenpair provides a route back to real coordinates: the real and imaginary parts of an eigenvector form a real basis, and in that basis the matrix has a simple rotation-scaling form. Once the form is known, selected powers can be computed using powers of a complex number. We work over \(\mathbb{R}\) for the matrices and use \(\mathbb{C}\) when discussing their eigenvalues and eigenvectors.

Suppose \(\mathbf{A}\in\mathbb{R}^{2\times 2}\) has eigenvalue \(\lambda=a+bi\), where \(a,b\in\mathbb{R}\) and \(b\ne0\), and let \(\mathbf{z}=\mathbf{u}+i\mathbf{v}\ne\mathbf{0}\) be a corresponding eigenvector, with \(\mathbf{u},\mathbf{v}\in\mathbb{R}^2\). The eigenvector equation \(\mathbf{A}\mathbf{z}=\lambda\mathbf{z}\) is an equality of complex vectors. Expanding each side and equating real and imaginary parts gives \(\mathbf{A}\mathbf{u}=a\mathbf{u}-b\mathbf{v}\) and \(\mathbf{A}\mathbf{v}=b\mathbf{u}+a\mathbf{v}\). These two real equations determine the matrix in the basis formed from \(\mathbf{u}\) and \(\mathbf{v}\).

The sign of the angle follows from the displayed convention for the block. Some texts use the opposite sign in the off-diagonal entries and describe the resulting angle differently. Here, the block is fixed by the order \((\mathbf{u},\mathbf{v})\) of the real and imaginary parts.

Why the Real and Imaginary Parts Give a Basis

Proof. Equating the real and imaginary parts of \(\mathbf{A}\mathbf{z}=(a+bi)(\mathbf{u}+i\mathbf{v})\) gives \(\mathbf{A}\mathbf{u}=a\mathbf{u}-b\mathbf{v}\) and \(\mathbf{A}\mathbf{v}=b\mathbf{u}+a\mathbf{v}\). To prove independence, suppose \(s\mathbf{u}+t\mathbf{v}=\mathbf{0}\) for real \(s,t\), not both zero. Then \(\mathbf{z}=\mathbf{u}+i\mathbf{v}\) is a complex scalar multiple of a nonzero real vector: if \(s\ne0\), the relation gives \(\mathbf{u}=-(t/s)\mathbf{v}\), and \(\mathbf{v}\ne\mathbf{0}\), since otherwise \(\mathbf{z}=\mathbf{0}\); if \(s=0\), then \(t\ne0\), \(\mathbf{v}=\mathbf{0}\), and \(\mathbf{u}\ne\mathbf{0}\). In either case there are a nonzero real vector \(\mathbf{w}\) and a nonzero complex scalar \(c\) such that \(\mathbf{z}=c\mathbf{w}\). The eigenvector equation then implies \(\mathbf{A}\mathbf{w}=\lambda\mathbf{w}\), after dividing by \(c\). Its left side is real, while its right side has nonzero imaginary part \(b\mathbf{w}\), a contradiction. Thus \(\mathbf{u},\mathbf{v}\) are independent, so the \(2\times2\) matrix \(\mathbf{P}\) with these columns is invertible. The two real-part equations say \(\mathbf{A}\mathbf{P}=\begin{bmatrix}a\mathbf{u}-b\mathbf{v}&b\mathbf{u}+a\mathbf{v}\end{bmatrix}=\mathbf{P}\begin{bmatrix}a&b\\-b&a\end{bmatrix}\). Multiplying on the left by \(\mathbf{P}^{-1}\) proves the claimed form. \(\square\)

Because \(\mathbf{A}\) is real, conjugating the eigenvector equation also shows that \(\overline{\lambda}=a-bi\) is an eigenvalue with eigenvector \(\overline{\mathbf{z}}=\mathbf{u}-i\mathbf{v}\). Thus the pair of complex eigenvectors is consistent with the real basis above; it does not supply two additional independent real directions.

Powers of the Real Block

Proof. For any real pairs \((a,b)\) and \((c,d)\), matrix multiplication gives \(\mathbf{C}(a,b)\mathbf{C}(c,d)=\begin{bmatrix}ac-bd&ad+bc\\-(ad+bc)&ac-bd\end{bmatrix}\). The entries \(ac-bd\) and \(ad+bc\) are the real and imaginary parts of \((a+bi)(c+di)\). For \(n=0\), the claimed formula holds because \(\mathbf{C}(a,b)^0=\mathbf{I}_2=\mathbf{C}(1,0)\) and \(\lambda^0=1\). If it holds for \(n\), multiply its right side by \(\mathbf{C}(a,b)\); the multiplication identity just shown gives the block whose entries are the real and imaginary parts of \(\lambda^n\lambda=\lambda^{n+1}\). This proves the formula by induction. For the similarity statement, the case \(n=0\) is \(\mathbf{I}_2=\mathbf{P}\mathbf{I}_2\mathbf{P}^{-1}\). If it holds for \(n\), then \(\mathbf{A}^{n+1}=\mathbf{A}^n\mathbf{A}=\mathbf{P}\mathbf{C}(a,b)^n\mathbf{P}^{-1}\mathbf{P}\mathbf{C}(a,b)\mathbf{P}^{-1}=\mathbf{P}\mathbf{C}(a,b)^{n+1}\mathbf{P}^{-1}\), because \(\mathbf{P}^{-1}\mathbf{P}=\mathbf{I}_2\). Induction proves the second formula. \(\square\)

This result gives a practical calculation: raise the complex number \(a+bi\) to the desired power, place its real and imaginary parts in the block, then change back to the original basis. The direct matrix product remains a useful independent check.

A Nonstandard Real Basis

A matrix need not look like a rotation-scaling block in the standard coordinates. The real and imaginary parts of an eigenvector can supply a different basis, and the similarity formula then carries the block calculation back to the original coordinates.

Common Conjugation and Sign Errors

For real \(\mathbf{A}\), conjugating an eigenpair changes both the eigenvalue and the eigenvector. For the matrix \(\mathbf{B}\) above, \(\mathbf{z}=(1,-i)^\top\) belongs to \(2+i\), while \(\overline{\mathbf{z}}=(1,i)^\top\) belongs to \(2-i\). A tempting but wrong calculation assigns \((1,i)^\top\) to \(2+i\). In fact, \(\mathbf{B}\begin{bmatrix}1\\i\end{bmatrix}=\begin{bmatrix}2-i\\1+2i\end{bmatrix}\), whereas \((2+i)\begin{bmatrix}1\\i\end{bmatrix}=\begin{bmatrix}2+i\\-1+2i\end{bmatrix}\). The vectors are not equal, so that proposed eigenpair is incorrect. The correction is to conjugate the whole equation: \(\mathbf{B}\overline{\mathbf{z}}=\overline{2+i}\,\overline{\mathbf{z}}=(2-i)\overline{\mathbf{z}}\).

Also keep the column order consistent. The block \(\begin{bmatrix}a&b\\-b&a\end{bmatrix}\) follows from the basis \((\mathbf{u},\mathbf{v})\) and the equations \(\mathbf{A}\mathbf{u}=a\mathbf{u}-b\mathbf{v}\), \(\mathbf{A}\mathbf{v}=b\mathbf{u}+a\mathbf{v}\). Reversing the order or replacing \(\mathbf{v}\) by \(-\mathbf{v}\) changes the displayed signs. It does not change the underlying matrix or the eigenvalues, but it does change the similarity calculation; always verify \(\mathbf{A}\mathbf{P}=\mathbf{P}\mathbf{C}\) for the particular basis matrix used.

The approach is useful because it turns repeated multiplication of a real matrix into a power of one complex number, while retaining a real answer in the original coordinates. For a reliable calculation, identify the eigenvalue and eigenvector, form \(\mathbf{P}\) from the real and imaginary parts in the stated order, verify the similarity, compute the complex power exactly, and check the final matrix by direct multiplication when the exponent is small.