What equation gives the eigenvalues?

The characteristic equation \det(\mathbf{A}-\lambda\mathbf{I})=0.

Characteristic equation of a 2×2 matrix in terms of trace and determinant?

\lambda^2-(trace)\lambda+\det\mathbf{A}=0

How do you find an eigenvector for a given eigenvalue?

Solve (\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}.

Why are the equations for an eigenvector dependent?

Because \mathbf{A}-\lambda\mathbf{I} is singular, so there is a line (or plane) of solutions.

Is the zero vector an eigenvector?

No; eigenvectors are non-zero by definition.

If \mathbf{x} is an eigenvector, what else is?

Any non-zero scalar multiple of \mathbf{x}, with the same eigenvalue.

What is a normalised eigenvector?

An eigenvector divided by its magnitude, so that it has magnitude 1.

Normalise \begin{pmatrix} 3 \\ 4 \end{pmatrix}.

\frac15\begin{pmatrix} 3 \\ 4 \end{pmatrix}

What does an eigenvalue of 0 tell you?

The matrix is singular: \det\mathbf{A}=0.

What degree is the characteristic equation of an n × n matrix?

Degree n, so a 3 × 3 matrix gives a cubic.

Eigenvalues and eigenvectorsEdexcel International A Level Further Maths: Flashcards

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Define an eigenvector and eigenvalue of $\mathbf{A}$.

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Define an eigenvector and eigenvalue of $\mathbf{A}$ .: A non-zero $\mathbf{x}$ with $\mathbf{A}\mathbf{x}=\lambda\mathbf{x}$ ; $\lambda$ is the eigenvalue.
What equation gives the eigenvalues?: The characteristic equation $\det(\mathbf{A}-\lambda\mathbf{I})=0$ .
Characteristic equation of a 2×2 matrix in terms of trace and determinant?: $\lambda^2-(\text{trace})\lambda+\det\mathbf{A}=0$
Sum of the eigenvalues equals what?: The trace of the matrix.
Product of the eigenvalues equals what?: The determinant of the matrix.
How do you find an eigenvector for a given eigenvalue?: Solve $(\mathbf{A}-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}$ .
Why are the equations for an eigenvector dependent?: Because $\mathbf{A}-\lambda\mathbf{I}$ is singular, so there is a line (or plane) of solutions.
Is the zero vector an eigenvector?: No; eigenvectors are non-zero by definition.
If $\mathbf{x}$ is an eigenvector, what else is?: Any non-zero scalar multiple of $\mathbf{x}$ , with the same eigenvalue.
What is a normalised eigenvector?: An eigenvector divided by its magnitude, so that it has magnitude 1.
Normalise $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .: $\frac15\begin{pmatrix} 3 \\ 4 \end{pmatrix}$
What does an eigenvalue of 0 tell you?: The matrix is singular: $\det\mathbf{A}=0$ .
What degree is the characteristic equation of an n × n matrix?: Degree n, so a 3 × 3 matrix gives a cubic.

Exam questions on Eigenvalues and eigenvectors

$\mathbf{A}=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ .
Find a normalised eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $2$ .2 marks
$\mathbf{B}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end{pmatrix}$ .
Find an eigenvector of $\mathbf{B}$ corresponding to the eigenvalue $2$ .2 marks
The matrix $\mathbf{M}=\begin{pmatrix} 5 & k \\ 1 & 2 \end{pmatrix}$ , where $k$ is a constant, has $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ as an eigenvector.
Find the eigenvalue corresponding to this eigenvector, and the value of $k$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).