$2$-norm of a matrix &$ 2$-norm of a vector

Question

I have a notation issue when I am doing a homework.

I am asked to minimize $||Ax - b||_2^2 + \alpha||x||_2^2$.

Here, A is a $4\times5$ matrix while b is a $4 \times 1$ column vector.

I suppose Ax - b is also a $4 \times 1$ column vector

Does $||Ax-b||_2$ refers to the 2-norm of a matrix or the 2 norm of vector?

Another thing to clarify:

Is the $2$-norm of a matrix equal to the square of its dominant eigenvalue?

If A is a matrix, is $||A||_2 = \sqrt{\lambda_{max}}$ ?

In that case, is $||A||_2^2$ = $\lambda_{max}$?

If A is a vector, is $||A||_2$ = $[\sum|x_{i}|^2]^{\frac{1}{2}}$ ?

In that case, is $||A||_2$ = $[\sum|x_{i}|^2]$

Thank you.

Answer 1

You need to understand that $Ax-b$ is a vector hence I would expect it to refer to the 2 norm of that vector. As for the other question:

\begin{align} ||A||_2 = \sup_{x \neq0} \frac{||Ax||_2}{||x||_2} = \sqrt{\lambda_{max}(A^TA)} \end{align}

Ill leave it open to you to explicitly show this, but think about the fact that without loss of generality, we can assume $||x||=1$ and hence we want to find the vector $x$ that maximises $||Ax||_2$. Now decompose $x$ into a linear combination of its eigenvectors (assume it has a full set), and think what combination would maximise the above expression.