How Do I Calculate Eigenvectors?

To calculate eigenvectors, you first need to find the corresponding eigenvalues of a square matrix. The process involves two main steps: solving the characteristic equation to find the eigenvalues and then solving a linear system for each eigenvalue to find the eigenvectors.

Here is a breakdown of the process:

Steps to Calculate Eigenvectors

Calculating eigenvectors requires finding the eigenvalues first. According to the reference, the steps are as follows:

Step 1: Find the Eigenvalues (λ)

The first crucial step is to find the eigenvalues of the given square matrix, A. This is done by solving the characteristic equation |A - λI| = 0.

• A: The square matrix for which you want to find eigenvectors.
• λ: A scalar value representing an eigenvalue.
• I: The identity matrix of the same dimension as A.
• |A - λI|: The determinant of the matrix (A - λI).
• 0: The zero vector or matrix.

Solving this determinant equation results in a polynomial in terms of λ. The roots of this polynomial are the eigenvalues of matrix A.

Step 2: Find the Eigenvectors (v) for Each Eigenvalue

Once you have found the eigenvalues (λ₁ , λ₂ , etc.), you need to find the corresponding eigenvectors for each eigenvalue. For each value of λ found in Step 1, substitute it back into the equation (A - λI) v = O and solve for the non-zero vector v.

• (A - λI): The matrix obtained by subtracting λ times the identity matrix from A.
• v: The eigenvector you are trying to find. This is a non-zero column vector.
• O: The zero vector.

The solutions v to this system of linear equations, excluding the trivial solution v = O, are the eigenvectors corresponding to that specific eigenvalue λ. Eigenvectors are typically expressed in their simplest form (e.g., scaled by a constant).

Illustrative Example

Let's consider a simple 2x2 matrix to demonstrate the process:

Let $A = \begin{pmatrix} 3 & 1 \ 1 & 3 \end{pmatrix}$.

Step 1: Find Eigenvalues

Calculate $A - λI$:
$A - λI = \begin{pmatrix} 3 & 1 \ 1 & 3 \end{pmatrix} - λ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 3-λ & 1 \ 1 & 3-λ \end{pmatrix}$

Solve the characteristic equation $|A - λI| = 0$:
$|A - λI| = (3-λ)(3-λ) - (1)(1) = (3-λ)^2 - 1 = 0$
$9 - 6λ + λ^2 - 1 = 0$
$λ^2 - 6λ + 8 = 0$
Factoring the quadratic equation:
$(λ - 2)(λ - 4) = 0$
The eigenvalues are $λ₁ = 2$ and $λ₂ = 4$.

Step 2: Find Eigenvectors

For λ₁ = 2:
Solve $(A - 2I) v = O$:
$(A - 2I) = \begin{pmatrix} 3-2 & 1 \ 1 & 3-2 \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix}$
We need to solve $\begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix} \begin{pmatrix} v₁ \ v₂ \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}$.
This gives the equations:
$1v₁ + 1v₂ = 0 \implies v₁ = -v₂$
$1v₁ + 1v₂ = 0 \implies v₁ = -v₂$
Let $v₂ = t$, where $t$ is any non-zero scalar. Then $v₁ = -t$.
The eigenvectors for $λ₁ = 2$ are of the form $\begin{pmatrix} -t \ t \end{pmatrix} = t \begin{pmatrix} -1 \ 1 \end{pmatrix}$.
A common choice for the eigenvector is $\begin{pmatrix} -1 \ 1 \end{pmatrix}$ (when $t=1$) or $\begin{pmatrix} 1 \ -1 \end{pmatrix}$ (when $t=-1$).

For λ₂ = 4:
Solve $(A - 4I) v = O$:
$(A - 4I) = \begin{pmatrix} 3-4 & 1 \ 1 & 3-4 \end{pmatrix} = \begin{pmatrix} -1 & 1 \ 1 & -1 \end{pmatrix}$
We need to solve $\begin{pmatrix} -1 & 1 \ 1 & -1 \end{pmatrix} \begin{pmatrix} v₁ \ v₂ \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}$.
This gives the equations:
$-1v₁ + 1v₂ = 0 \implies v₁ = v₂$
$1v₁ - 1v₂ = 0 \implies v₁ = v₂$
Let $v₂ = t$, where $t$ is any non-zero scalar. Then $v₁ = t$.
The eigenvectors for $λ₂ = 4$ are of the form $\begin{pmatrix} t \ t \end{pmatrix} = t \begin{pmatrix} 1 \ 1 \end{pmatrix}$.
A common choice for the eigenvector is $\begin{pmatrix} 1 \ 1 \end{pmatrix}$ (when $t=1$).

Summary Table

Understanding how to calculate eigenvectors is fundamental in linear algebra and has applications in various fields like physics, engineering, and computer science.