What is the characteristic equation?

\det(\mathbf{A}-\lambda\mathbf{I})=0

Characteristic equation of a 2\times2 matrix in terms of trace and determinant?

\lambda^2-(trace)\lambda+\det=0

What do the eigenvalues of a matrix sum and multiply to?

Sum = trace; product = determinant.

How do you find an eigenvector?

Solve (\mathbf{A}-\lambda\mathbf{I})\mathbf{v}=\mathbf{0} for each eigenvalue \lambda.

What is a normalised eigenvector?

An eigenvector divided by its magnitude, so its length is 1.

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

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

What happens when the characteristic equation has a repeated root?

The eigenvalue has multiplicity 2 and may give only one eigenvector direction.

How do complex eigenvalues of a real matrix arise?

In conjugate pairs p\pm qi, with conjugate eigenvectors.

What does an eigenvector mean geometrically?

A direction through the origin that the transformation maps onto itself (an invariant line), scaled by \lambda.

Eigenvalues and eigenvectorsEdexcel A-Level Further Maths: Flashcards

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Define an eigenvector and eigenvalue.

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Define an eigenvector and eigenvalue.: $\mathbf{A}\mathbf{v}=\lambda\mathbf{v}$ with $\mathbf{v}\neq\mathbf{0}$ ; $\mathbf{v}$ is the eigenvector, $\lambda$ the eigenvalue.
What is the characteristic equation?: $\det(\mathbf{A}-\lambda\mathbf{I})=0$
Characteristic equation of a $2\times2$ matrix in terms of trace and determinant?: $\lambda^2-(\text{trace})\lambda+\det=0$
What do the eigenvalues of a matrix sum and multiply to?: Sum $=$ trace; product $=$ determinant.
How do you find an eigenvector?: Solve $(\mathbf{A}-\lambda\mathbf{I})\mathbf{v}=\mathbf{0}$ for each eigenvalue $\lambda$ .
Eigenvalues of $\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$ ?: $1$ and $4$
What is a normalised eigenvector?: An eigenvector divided by its magnitude, so its length is $1$ .
Normalise $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .: $\frac15\begin{pmatrix} 3 \\ 4 \end{pmatrix}$
What happens when the characteristic equation has a repeated root?: The eigenvalue has multiplicity 2 and may give only one eigenvector direction.
How do complex eigenvalues of a real matrix arise?: In conjugate pairs $p\pm qi$ , with conjugate eigenvectors.
If $\mathbf{A}\mathbf{v}=\lambda\mathbf{v}$ , what is $\mathbf{A}^n\mathbf{v}$ ?: $\lambda^n\mathbf{v}$
What does an eigenvector mean geometrically?: A direction through the origin that the transformation maps onto itself (an invariant line), scaled by $\lambda$ .

Exam questions on Eigenvalues and eigenvectors

The matrix $\mathbf{A}=\begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$ .
Find an eigenvector of $\mathbf{A}$ corresponding to the eigenvalue $1$ .2 marks
The matrix $\mathbf{B}=\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}$ .
Find an eigenvector of $\mathbf{B}$ corresponding to the eigenvalue $1-2i$ .2 marks
The matrix $\mathbf{C}=\begin{pmatrix} 3 & -1 \\ 1 & 1 \end{pmatrix}$ .
Show that $\mathbf{C}$ has a repeated eigenvalue and find its value.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).