State the diagonalisation form of a matrix \(A\).

What must the columns of \(P\) be in \(A=PDP^{-1}\)?

If \(P=[\mathbf v_1\ \mathbf v_2]\) and \(A\mathbf v_1=4\mathbf v_1\), \(A\mathbf v_2=-1\mathbf v_2\), write \(D\).

Is the diagonal matrix \(A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\) already diagonalised?

Given eigenvectors \(\mathbf v_1=(1,0)\), \(\mathbf v_2=(1,1)\) with eigenvalues \(2\), \(3\), build \(P\) and \(D\).

For \(P=\begin{pmatrix}1&1\\0&1\end{pmatrix}\) and \(D=\begin{pmatrix}2&0\\0&3\end{pmatrix}\), compute \(PD\).

Verify \(AP=PD\) for \(A=\begin{pmatrix}2&1\\0&3\end{pmatrix}\), \(P=\begin{pmatrix}1&1\\0&1\end{pmatrix}\), \(D=\begin{pmatrix}2&0\\0&3\end{pmatrix}\).

Why does the order of columns in \(P\) matter?

Diagonalise \(A=\begin{pmatrix}2&0\\0&5\end{pmatrix}\).

Diagonalise \(A=\begin{pmatrix}2&1\\0&3\end{pmatrix}\).

Explain why \(n\) distinct eigenvalues guarantee diagonalisation of an \(n\times n\) matrix.

Explain why \(A=PDP^{-1}\) means the map is simple in eigenvector coordinates.

Show that \(A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\) is not diagonalizable.

For which \(k\) is \(A=\begin{pmatrix}2&k\\0&2\end{pmatrix}\) diagonalizable?

Diagonalise \(A=\begin{pmatrix}4&0\\0&4\end{pmatrix}\) in two different ways.

A diagonalisation uses \(P=\begin{pmatrix}1&1\\1&-1\end{pmatrix}\) and \(D=\begin{pmatrix}6&0\\0&2\end{pmatrix}\). What are two eigenvector-eigenvalue pairs?

Find \(A\) from \(P=\begin{pmatrix}1&1\\0&1\end{pmatrix}\), \(D=\begin{pmatrix}2&0\\0&3\end{pmatrix}\), and \(A=PDP^{-1}\).

A student builds \(P\) from two eigenvectors that are scalar multiples. Explain why this cannot diagonalise a \(2\times2\) matrix.

Prove that \(AP=PD\) when \(P\) has eigenvectors as columns and \(D\) has matching eigenvalues.

Explain why repeated eigenvalues require checking eigenspace dimension before diagonalising.