Ytukyg

This seems to be a very easy exercise, but as I never had previous contact with vector induced norm of a matrix, I am getting stuck...

Given the matrix:
$$A=
begin{bmatrix}
2 & 0\
0 & 1\
end{bmatrix}
$$

I am supposed to compute its vector-induced $|A|_2$ norm, defined as:

$$|A|_2=sup_{vneq 0}frac{|Av|_2}{|v|_2}$$

My attempt so far was:

Define a vector $v=(v_1,v_2)$ so that I end up with:

$$|A|_2=sup_{vneq 0}frac{|Av|_2}{|v|_2}=sup_{vneq 0}sqrt{frac{4v_{1}^{2}+v_{2}^{2}}{v_{1}^{2}+v_{2}^{2}}}$$

It seems reasonable to choose a vector $v$ such that $|v|_2=1$, then this would result in a problem where I want to find:

$$|A|_2=sup_{|v|_2=1}sqrt{4v_{1}^{2}+v_{2}^{2}}$$

Which kind of seems like an optimization problem where the constraint is:

$$sqrt{v_{1}^{2}+v_{2}^{2}}=1$$

And this is where I get stuck!

Is this a correct way to solve this? If yes, could you help me finish this line of thought? Would Lagrange Multiplier do the job?

I really would like to solve it "the hard way" before using other properties/definitions such as:

$$|A|_2=sigma_{max}(A)$$

Thanks in advance!

Alternative:

You want to solve $$max 4v_1^2 + v_2^2$$

subject to $$v_1^2+v_2^2=1$$

Let $y_i = v_i^2$, we have

$$max 4y_1+y_2$$

subject to $$y_1+y_2=1$$
$$y_i ge 0 ,forall i in {1,2}$$

which is just a linear programming problem which has optimal solution at the corners of the polyhedral.

We want $y_1$ to be large. Hence the optimal $4y_1+y_2=4(1)+0=4$.

Taking square root gives us $2$ as the optimal solution

Here's one way to find it for a particular matrix such as the one in your example. Based on the work you've already done, observe that
$$|Av|_2^2=4v_1^2+v_2^2le4v_1^2+4v_2^2=4|v|_2^2 quad implies quad |A|_2le2.$$
Now if you can find a specific vector $v$ such that this value of $2$ is attained, i.e. such that $|Av|=2$ for a vector $v$ with $|v|_2=1$, then you will have demonstrated that $|A|_2=2$. And finding such a vector has a lot to do with eigenvectors….

Given a matrix $A$ the singular values of $A$ are just the square roots of the eigenvalues of the matrix $A^*A$. The largest singular value is the operator norm which is the quantity you want to compute.