Matrix norm in Banach space

[kalish1]

How can I calculate the following matrix norm in a Banach Space:

$$
A=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix}
?$$

I have tried $$\|A\|=\sup\limits_{\|x\|=1}\|Az\|$$

and then did $$Az=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5x-2y \\ x-y \end{pmatrix}.$$

Now, how can I maximize the function $\|Az\|$? I'm not sure.

The norm is not specified in the problem. Should I just go with the square root of the sum of the squares of the compoments of Az, i.e. maximize $\sqrt{(5x-2y)^2+(x-y)^2}$ given the constraint $x^2+y^2=1$? Or should I maximize $\{{(5x-2y)^p+(x-y)^p}\}^{1/p}$ given the constraint $x^2+y^2=1$?

Any help would be appreciated.

I have crossposted this question here: calculus - Matrix norm in Banach space - Mathematics Stack Exchange

 

[stainburg]

If you want to calculate 2-norm,

(a) easy way I: use software such as Matlab;)

(b) easy way II: use \(\displaystyle \sqrt{\text{sup}(\lambda)}\), where \(\displaystyle \lambda\) is eigenvalue of \(\displaystyle A^TA\).

(c) hard way: Let \(\displaystyle z=[x\text{ }y]^T\) be unit vector. You need to maximize \(\displaystyle \parallel Az\parallel_2^2=z^TA^TAz\), subject to \(\displaystyle \parallel z\parallel_2^2=z^Tz=1\). By Lagrange multiplier method, it is to solve equations \(\displaystyle \bigtriangledown_z (z^TA^TAz-\lambda (z^Tz-1))=0\) and \(\displaystyle z^Tz=1\) (actually, it's not a typical optimization problem, but we can still do it like this, why?). At last, we have \(\displaystyle A^TAz=\lambda z\). \(\displaystyle \lambda\) is what easy way II speaks of.

 

[Fermat1]

How can I calculate the following matrix norm in a Banach Space:

$$
A=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix}
?$$

I have tried $$\|A\|=\sup\limits_{\|x\|=1}\|Az\|$$

and then did $$Az=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5x-2y \\ x-y \end{pmatrix}.$$

Now, how can I maximize the function $\|Az\|$? I'm not sure.

The norm is not specified in the problem. Should I just go with the square root of the sum of the squares of the compoments of Az, i.e. maximize $\sqrt{(5x-2y)^2+(x-y)^2}$ given the constraint $x^2+y^2=1$? Or should I maximize $\{{(5x-2y)^p+(x-y)^p}\}^{1/p}$ given the constraint $x^2+y^2=1$?

Any help would be appreciated.

I have crossposted this question here: calculus - Matrix norm in Banach space - Mathematics Stack Exchange

Since you are taking the sup of non-negative numbers, we may as well maximise $(5x-2y)^2+(x-y)^2$ subject to $x^2+y^2=1$ and then take square root at the end. The condition implies $x=cos{\alpha}$ and $y=sin{\alpha}$ for some ${\alpha}$. Rewriting $(5x-2y)^2+(x-y)^2$ using that, its obvious that the max occurs when $cos{\alpha}$ is positive and $sin{\alpha}=-cos{\alpha}$. So $(5x-2y)^2+(x-y)^2=6.27$ to 3.s.f. Square rooting gives the norm as approximately 2.5.

 

[Opalg]

How can I calculate the following matrix norm in a Banach Space:

$$
A=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix}
?$$

I have tried $$\|A\|=\sup\limits_{\|x\|=1}\|Az\|$$

and then did $$Az=\begin{pmatrix}
5 & -2 \\
1 & -1 \\
\end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5x-2y \\ x-y \end{pmatrix}.$$

Now, how can I maximize the function $\|Az\|$? I'm not sure.

The norm is not specified in the problem. Should I just go with the square root of the sum of the squares of the compoments of Az, i.e. maximize $\sqrt{(5x-2y)^2+(x-y)^2}$ given the constraint $x^2+y^2=1$? Or should I maximize $\{{(5x-2y)^p+(x-y)^p}\}^{1/p}$ given the constraint $x^2+y^2=1$?

Any help would be appreciated.

I have crossposted this question here: calculus - Matrix norm in Banach space - Mathematics Stack Exchange

There are many different norms for a matrix considered as an operator on a Banach space. If you want the operator norm of $A$ as an operator on 2-dimensional Euclidean space then the easiest way to calculate it is by using the C*-identity $\|A\|^2 = \|A^*A\|$, where $A^*$ is the Hermitian adjoint of $A$. In this case, $A^*A = \begin{bmatrix}5&1 \\-2&-1 \end{bmatrix} \begin{bmatrix}5&-2 \\1&-1 \end{bmatrix} = \begin{bmatrix}26&-11 \\-11&5 \end{bmatrix}.$ Since $A^*A$ is a positive matrix, its norm will be its largest eigenvalue, which you can find in the usual way by solving the quadratic equation $\det(A^*A - \lambda I) = 0.$ Then take the square root to get $\|A\|.$

Edit. I hadn't noticed that stainburg already suggested this method.

 

[blue_raver22]

I can provide some guidance on how to approach this problem. First, it is important to understand the concept of a matrix norm in a Banach space. A matrix norm is a way to measure the size or magnitude of a matrix in a vector space. In a Banach space, the matrix norm is defined as the supremum (or maximum) of the norm of all possible vectors that can be obtained by multiplying the matrix with a unit vector. In simpler terms, it is the maximum value that the matrix can "stretch" a unit vector to.

Now, to calculate the matrix norm of the given matrix A, we can follow the steps provided in the question. First, we need to find the norm of Az, which is the result of multiplying A with a unit vector. As mentioned in the question, this can be done by finding the square root of the sum of squares of the components of Az. However, this is not the only way to define a matrix norm. Other common norms include the maximum absolute value of the components of Az and the sum of absolute values of the components of Az.

Once we have the norm of Az, we need to find the maximum value of this norm for all possible unit vectors. In this case, since the unit vectors are constrained to have a norm of 1, we can use techniques from calculus to find the maximum value. This involves setting up and solving an optimization problem, where we maximize the norm of Az subject to the constraint that the norm of the unit vector is 1. This can be done by finding the critical points of the function and evaluating them at the endpoints of the constraint.

In summary, to calculate the matrix norm of A in a Banach space, we need to find the norm of Az (the result of multiplying A with a unit vector) and then maximize this value over all possible unit vectors. This can be done by setting up and solving an optimization problem. The specific norm used in the calculation will depend on the definition given in the problem.