2.7. Eigenvalues and Eigenvectors

Example 2.7.1 (Fibonacci Numbers). We can use a change of basis to create a formula for the Fibonacci numbers.

$$ a_n = \frac{5+\sqrt{5}}{10} \bigg(\frac{1+\sqrt{5}}{2}\bigg)^n + \frac{5-\sqrt{5}}{10}\bigg(\frac{1-\sqrt{5}}{2}\bigg)^n. $$

For any matrix $A$, the eigenvalues are the roots of the characteristic polynomial $\det (\lambda I - A) = 0$. This is not a good algorithm in practice because roots of high degree polynomials are hard to compute. In practice, QR decomposition is used. For symmetric matrices, the Jacobi algorithm is used.

Definition 2.7.2 (Eigenvector, eigenvalue, multiplicity). Let $V$ be a complex vector spce and $T: V\to V$ a linear transformation. A nonzero vector $v$ such that

$$ Tv = \lambda v $$

for some number $\lambda$ is called an eigenvector of $T$. The number $\lambda$ is the corresponding eigenvalue. The multiplicity of an eigenvalue $\lambda$ is the dimension of the eigenspace $\{v\mid Tv = \lambda v\}$.

Definition 2.7.3 (Eigenbasis). A basis for a complex vector space $V$ is an eigenbasis of $V$ for a linear transformation $T$ if each element of the basis is an eigenvector of $T$.

Diagonalization and eigenvectors

Proposition 2.7.5 (Diagonalization and eigenvectors). Let $A$ be an $n\times n$ matrix and $P = [\vec v_1 , \dots, \vec v_n]$ an invertible $n\times n$ matrix.

The eigenvalues of $A$ and the eigenvalues of $P^{-1}AP$ coincide.
If $(\vec v_1, \dots, \vec v_n)$ is an eigenbasis of $\Complex^n$ for $A$, with $A\vec v_i = \lambda_i \vec v_i$, then $P^{-1}AP$ is a diagonal matrix with diagonal entires $\lambda_1, \dots, \lambda_n$.
Conversely, if $P^{-1}AP$ is diagonal with diagonal entries $\lambda_i$, the columns of $P$ are eigenvectors of $A$ with eigenvalues $\lambda_i$.

<aside> 💡 $P^{-1}AP$ should remind you of the change of base formula.

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Finding an eigenbasis

Suppose we have the vectors $\vec e_1, A\vec e_1, A^2 \vec e_1, \dots$ What is the first $A^k \vec e_1$ which is linearly dependent on the earlier vectors? This is just what row reduction is for!

$$ A^k \vec e_1 = a_0 \vec e_1 + a_1A\vec e_2 +a_2A^2\vec e_2 + \dots $$

The roots of the polynomial $A^k - a_{k-1}A^{k-1} - \dots - a_0 = 0$ are eigenvalues of $A$.