Result of \mathbf{P}^{-1}\mathbf{A}\mathbf{P}?

The diagonal matrix \mathbf{D} of eigenvalues, in the same order as the columns of \mathbf{P}.

Formula for \mathbf{A}^n using diagonalisation?

\mathbf{A}^n=\mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}

When can a 2\times2 matrix not be diagonalised?

When it lacks two independent eigenvectors, for example a repeated eigenvalue with one eigenvector direction.

Two properties of a real symmetric matrix's eigenvalues and eigenvectors?

Real eigenvalues; eigenvectors for distinct eigenvalues are perpendicular.

What is an orthogonal matrix?

\mathbf{Q}^{T}\mathbf{Q}=\mathbf{I}, so \mathbf{Q}^{-1}=\mathbf{Q}^{T}; columns are perpendicular unit vectors.

Orthogonal diagonalisation of a symmetric matrix?

\mathbf{Q}^{T}\mathbf{S}\mathbf{Q}=\mathbf{D} with \mathbf{Q} made of normalised eigenvectors.

State the Cayley-Hamilton theorem for a 2\times2 matrix.

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

How does Cayley-Hamilton give an inverse?

Rearrange to \mathbf{A}(\ldots)=k\mathbf{I}, so \mathbf{A}^{-1}=\frac1k(\ldots).

Diagonalisation and the Cayley-Hamilton theoremEdexcel A-Level Further Maths: Flashcards

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How do you build $\mathbf{P}$ to diagonalise $\mathbf{A}$ ?: Put the eigenvectors of $\mathbf{A}$ as columns of $\mathbf{P}$ .
Result of $\mathbf{P}^{-1}\mathbf{A}\mathbf{P}$ ?: The diagonal matrix $\mathbf{D}$ of eigenvalues, in the same order as the columns of $\mathbf{P}$ .
If you swap the columns of $\mathbf{P}$ , what changes in $\mathbf{D}$ ?: The diagonal entries swap too.
Formula for $\mathbf{A}^n$ using diagonalisation?: $\mathbf{A}^n=\mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}$
What is $\mathbf{D}^n$ for $\mathbf{D}=\mathrm{diag}(\lambda_1,\lambda_2)$ ?: $\mathrm{diag}(\lambda_1^n,\lambda_2^n)$
When can a $2\times2$ matrix not be diagonalised?: When it lacks two independent eigenvectors, for example a repeated eigenvalue with one eigenvector direction.
What is a symmetric matrix?: $\mathbf{S}=\mathbf{S}^{\mathrm{T}}$
Two properties of a real symmetric matrix's eigenvalues and eigenvectors?: Real eigenvalues; eigenvectors for distinct eigenvalues are perpendicular.
What is an orthogonal matrix?: $\mathbf{Q}^{\mathrm{T}}\mathbf{Q}=\mathbf{I}$ , so $\mathbf{Q}^{-1}=\mathbf{Q}^{\mathrm{T}}$ ; columns are perpendicular unit vectors.
Orthogonal diagonalisation of a symmetric matrix?: $\mathbf{Q}^{\mathrm{T}}\mathbf{S}\mathbf{Q}=\mathbf{D}$ with $\mathbf{Q}$ made of normalised eigenvectors.
State the Cayley-Hamilton theorem for a $2\times2$ matrix.: $\mathbf{A}^2-(\text{trace})\mathbf{A}+(\det)\mathbf{I}=\mathbf{0}$
How does Cayley-Hamilton give an inverse?: Rearrange to $\mathbf{A}(\ldots)=k\mathbf{I}$ , so $\mathbf{A}^{-1}=\frac1k(\ldots)$ .

Exam questions on Diagonalisation and the Cayley-Hamilton theorem

The matrix $\mathbf{A}=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ has eigenvalues $2$ and $5$ , with eigenvectors $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ respectively. Let $\mathbf{P}=\begin{pmatrix} 1 & 1 \\ -2 & 1 \end{pmatrix}$ .
Write down a matrix $\mathbf{Q}$ such that $\mathbf{Q}^{-1}\mathbf{A}\mathbf{Q}=\begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}$ .2 marks
The symmetric matrix $\mathbf{S}=\begin{pmatrix} 5 & 2 \\ 2 & 2 \end{pmatrix}$ .
Find an orthogonal matrix $\mathbf{Q}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{Q}^{\mathrm{T}}\mathbf{S}\mathbf{Q}=\mathbf{D}$ .2 marks
The matrix $\mathbf{C}=\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ .
Find the characteristic equation of $\mathbf{C}$ and hence show that $\mathbf{C}^2=6\mathbf{C}-5\mathbf{I}$ .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).