32. Eigenvalues and Eigenvectors

Dated: 15-06-2025

Definition

A complex scalar \(\lambda\) satisfies

\[\det(A - \lambda I) = 0\]

If and only if there is a non zero vector¹ \(\vec x\) in \(\mathbb C^n\) such that

\[A \vec x = \lambda \vec x\]

We call \(\lambda\) a complex eigen value and \(\vec x\) a complex eigen vector corresponding to \(\lambda\).

Example

\[ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

The linear transformation² \(\vec x \to A \vec x\) on \(\mathbb R^2\) rotates the plane counter clockwise through a quarter turn.

Solution

The action of \(A\) is periodic and a vector¹ is back where it started from. Obviously, no nonzero vector¹ is mapped into a multiple of itself, so \(A\) has no eigen vectors in \(\mathbb R^2\) and no real eigen values.
The characteristic equation³ of \(A\) is

\[\lambda^2 + 1 = 0\]

\[\implies \lambda = \pm \sqrt{-1} = \pm \iota\]

\[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ -\iota \end{bmatrix} = \begin{bmatrix} \iota \\ 1 \end{bmatrix} = \iota \begin{bmatrix} 1 \\ -\iota \end{bmatrix} \]

\[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ \iota \end{bmatrix} = \begin{bmatrix} -\iota \\ 1 \end{bmatrix} = -\iota \begin{bmatrix} 1 \\ \iota \end{bmatrix} \]

Thus \(\iota\) and \(- \iota\) are eigen values with \(\begin{bmatrix}1\\ -\iota\end{bmatrix}\) and \(\begin{bmatrix}1\\\iota\end{bmatrix}\) are corresponding eigen vectors.

Real and Imaginary Parts of Vectors¹

The real and imaginary parts of a complex vector \(x\) are the vectors¹ \(\textbf{Re } \vec x\) and \(\textbf{Im } \vec x\) formed from the real and imaginary parts of the entries of \(x\).

Example

\[ \vec{x} = \begin{bmatrix} 3-i \\ i \\ 2+5i \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} + i \begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix} \]

\[ \textbf{Re } \vec x = \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}, \quad \textbf{Im } \vec x = \begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix}, \]

\[ \bar{\vec{x}} = \begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix} - i \begin{bmatrix} -1 \\ 1 \\ 5 \end{bmatrix} = \begin{bmatrix} 3+i \\ -i \\ 2-5i \end{bmatrix} \]

Eigenvalues And Eigenvectors of a Real Matrix¹ that Acts on \(\mathbb C^n\)

If \(A\) is an \(n \times n\) matrix¹ whose entries are real then \(\overline{A \vec x} = \bar A \bar {\vec x} = A \bar {\vec x}\).
If \(\lambda\) is an eigen value of \(A\) with \(\vec x\) a corresponding vector¹ in \(\mathbb C^n\) then \(A \bar {\vec x} = \overline{A \vec x} = \overline{\lambda \vec x} = \bar \lambda \bar {\vec x}\).

Example

If \(\mathbb C = \begin{bmatrix}a & -b\\ b & a\end{bmatrix}\) where \(a, b \in \mathbb R \ \land a, b\ne 0\) then the eigen values of \(\mathbb C\) are \(\lambda = a \pm b \iota\).
Also, if \(r = |\lambda| = \sqrt{a^2 + b^2}\), then

\[ \mathbb C = r \begin{bmatrix} \frac a r & - \frac b r \\ \frac b r & \frac a r \end{bmatrix} = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix} \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \]

The transformation ² \(\vec x \to C \vec x\) can be viewed as composition of rotation through \(\theta\) and scaling by \(r = |\lambda|\).

Theorem

\[A = PCP^{-1}\]

\[ C = \begin{bmatrix} a & -b\\ b & a \end{bmatrix} \]

\[P = \begin{bmatrix}\textbf{Re } \vec v & \textbf{Im } \vec v\end{bmatrix} \quad \text{where } \lambda = a \pm b \iota, \ \vec v \in \mathbb C\]

References

Read more about notations and symbols.

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1. Read more about vectors. ↩↩↩↩↩↩↩↩

2. Read more about linear transformations. ↩↩

3. Read more about characteristic equation. ↩