AHL 1.15—Eigenvalues and eigenvectors: IB DP Math AI Notes

What Are Eigenvalues and Eigenvectors?

Most matrix-vector multiplications change both the direction and magnitude of a vector. But for certain special vectors — called eigenvectors — multiplication by a matrix only scales the vector, leaving its direction unchanged (or exactly reversed).

Eigenvector

A non-zero vector $v$ is an eigenvector of a square matrix $A$ if multiplying $A$ by $v$ produces a scalar multiple of $v$ :
$A v = λ v$
The scalar $λ$ is the corresponding eigenvalue.

Eigenvalue

The scalar $λ$ such that $A v = λ v$ for some non-zero vector $v$ . The eigenvalue tells you by how much the eigenvector is stretched, compressed, or reflected.

Key observations:

If $λ > 1$ : the eigenvector is stretched
If $0 < λ < 1$ : the eigenvector is compressed
If $λ < 0$ : the eigenvector is reversed in direction and scaled
If $λ = 1$ : the eigenvector is unchanged by the transformation
If $λ = 0$ : the eigenvector is mapped to the zero vector by $A$ (note: $A$ is singular in this case, and the eigenvector lies in the null space of $A$ )

Warning

An eigenvector must be non-zero by definition. The zero vector is excluded because $A 0 = λ 0$ holds trivially for any $λ$ , which is not useful or meaningful.

Note

Syllabus scope: For IB AI HL (AHL 1.15), you are only required to find eigenvalues and eigenvectors for $2 \times 2$ matrices. All examples and exam questions will involve $2 \times 2$ matrices unless otherwise stated.

What Are Eigenvalues and Eigenvectors?

The Characteristic Polynomial

To find eigenvalues, we rearrange the equation $A v = λ v$ :
$A v - λ v = 0 ⟹ (A - λ I) v = 0$

For this to have a non-zero solution $v$ , the matrix $(A - λ I)$ must be singular (non-invertible), which means its determinant must be zero:
$det (A - λ I) = 0$

This equation is called the characteristic equation, and the expression $det (A - λ I)$ is the characteristic polynomial.

Characteristic Polynomial

For a $2 \times 2$ matrix $A = (a c b d)$ , the characteristic polynomial is:
$p (λ) = det (A - λ I) = λ^{2} - (a + d) λ + (a d - b c)$
Note that $(a + d)$ is the trace of $A$ and $(a d - b c)$ is the determinant of $A$ , so:
$p (λ) = λ^{2} - tr (A) \cdot λ + det (A)$

The eigenvalues are the roots of this quadratic, which can be found by factoring or using:
$λ = \frac{tr ( A ) \pm [ tr ( A ) ] ^{2} - 4 d e t ( A )}{2}$

Exam Tip

Memorise the pattern: characteristic polynomial = $λ^{2} - (trace) λ + (det)$ . This saves time setting up the determinant calculation from scratch every time.

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What Are Eigenvalues and Eigenvectors?

The Characteristic Polynomial

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