Calculate the spectral norm

Question

Consider the four vectors $v_1, v_2, u_1, u_2 \in \mathbb{C}^2$ with

$$v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \qquad u_1 = \begin{pmatrix} 1 \\ i \end{pmatrix}, \qquad u_2 = \begin{pmatrix} 2+i \\ 1 \end{pmatrix}$$

Find the spectral norm of $\phi: \mathbb{C}^2 \to \mathbb{C}^2$ that maps the vector $v_i$ to the vector $u_i$.

Before people downvote: I know that the spectral norm is given as the square root of the max eigenvalue of the matrix $B$ with $B=A^{H}A$ where $A$ is the matrix representation of $\phi$. However, I struggle with the format of the task given as there is no matrix given and the vectors are not orthonormal.

Answer 1

Hint:

You have to find the images of the canonical basis $(e_1,e_2)$ knowing that $$\begin{cases}\phi(v_1)=\phi(e_1)+\phi(e_2)=\begin{bmatrix}1\\i\end{bmatrix}\\[1ex] \phi(v_2)=2\phi(e_1)+\phi(e_2)=\begin{bmatrix}2+i\\1\end{bmatrix}\end{cases}.$$ A simple elimination shows \begin{align*} \phi(e_1)&=\phi(v_2)-\phi(v_1)=\begin{bmatrix}1+i\\1-i\end{bmatrix},\\[1ex] \phi(e_2)&=2\phi(v_1)-\phi(v_2)=\begin{bmatrix}-i\\2i-1\end{bmatrix}. \end{align*}

Answer 2

We have two linear systems in $\mathrm A \in \mathbb C^{2 \times 2}$

$$\mathrm A \mathrm v_1 = \mathrm u_1 \qquad \qquad \qquad \mathrm A \mathrm v_2 = \mathrm u_2$$

Vectorizing, we obtain the following linear system of $4$ equations in $4$ unknowns

$$\begin{bmatrix} \mathrm v_1^T \otimes \mathrm I_2\\ \mathrm v_2^T \otimes \mathrm I_2\end{bmatrix} \mbox{vec} (\mathrm A) = \begin{bmatrix} \mathrm u_1 \\ \mathrm u_2\end{bmatrix}$$

Once we have $\mbox{vec} (\mathrm A)$, we un-vectorize it to obtain $\mathrm A$. Lastly, we compute the spectral norm $\|\mathrm A\|_2$.