If \(A=PDP^{-1}\), state a formula for \(A^n\).

If \(D=\operatorname{diag}(2,5)\), what is \(D^3\)?

If a mode has eigenvalue \(\lambda=\frac12\), what happens to that mode after many repeated updates?

If a mode has eigenvalue \(-2\), describe its repeated-update behaviour.

Compute \(D^4\) for \(D=\begin{pmatrix}3&0\\0&-1\end{pmatrix}\).

If \(\mathbf x_0=3\mathbf v_1-2\mathbf v_2\), with eigenvalues \(4\) and \(1\), find \(A^2\mathbf x_0\).

Given \(A=PDP^{-1}\), \(D=\operatorname{diag}(2,3)\), compute \(D^5\).

If \(P^{-1}\mathbf x_0=(2,-1)\) and \(D=\operatorname{diag}(3,4)\), find the eigenvector coordinates after two updates.

Show why \((PDP^{-1})^2=PD^2P^{-1}\).

If \(A\mathbf v_1=2\mathbf v_1\), \(A\mathbf v_2=5\mathbf v_2\), and \(\mathbf x=\mathbf v_1+3\mathbf v_2\), find \(A^3\mathbf x\).

Explain why diagonalisation is useful for repeated linear updates \(\mathbf x_{n+1}=A\mathbf x_n\).

Interpret eigenvalues \(0.8\), \(1\), and \(1.2\) in a repeated-update model.

Let \(A=PDP^{-1}\), with \(D=\operatorname{diag}(2,3)\), \(P=\begin{pmatrix}1&1\\0&1\end{pmatrix}\). Find \(A^2(1,1)\).

For \(D=\operatorname{diag}(-1,2)\), find all initial eigen-coordinates \((a,b)\) whose first component is unchanged after two updates.

A state has eigen-coordinates \((1,2)\) and eigenvalues \(\frac12\), \(-3\). Find its eigen-coordinates after \(n\) updates.

For which \(r\) does a mode with eigenvalue \(r\) decay under repeated updates?

A diagonalised system has eigenvalues \(2\) and \(3\). Which mode dominates long term if both initial mode amplitudes are non-zero?

A student computes \(A^n= P D P^{-1}\) instead of \(P D^n P^{-1}\). Explain the error.

Prove by cancellation that \(A^n=PD^nP^{-1}\) when \(A=PDP^{-1}\).

Explain why diagonalisation may fail as a method for powers even when eigenvalues are known.