# What Is an Example of a Matrix Norm?

• ericm1234
In summary, the conversation discusses the definition of the norm of an M x N matrix A, which measures how "long" A can make a unit vector. The norm is defined as the smallest real number that is the least upper bound for the set of all |Av| with |v|=1, or equivalently, as the radius of the smallest closed ball around 0 that contains A(S), where S is the unit sphere. In the case of finite-dimensional vector spaces, the norm is also related to the continuity of a linear operator.
ericm1234
My book goes on to say:
"If we consider both C^n and C^m with norms, then we define the norm of an M x N matrix A by.."
Then the formula says norm of A=sup (over abs(v)=1) of abs(Av) = sup (over v does not equal 0) abs(Av)/abs(v)
Can someone please provide me at least one example of what this means?

Suppose we take
$$A = \left[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}\right]$$
For simplicity, let's work in $\mathbb{R}^2$ and use the 2-norm (euclidean length). The unit vectors in $\mathbb{R}^2$ consist of all points on the unit circle, i.e., any vector of the form
$$v = \left[\begin{matrix} \cos(\theta) \\ \sin(\theta) \end{matrix} \right]$$
The norm of $A$ is obtained by maximizing $||Av||$ with respect to $\theta$, i.e., by choosing the unit-length $v$ in the direction that maximizes the length of $Av$. We can carry out this computation explicitly:
$$Av = \left[\begin{matrix} \cos(\theta) + 2\sin(\theta) \\ 3\cos(\theta) + 4\sin(\theta) \end{matrix} \right]$$
so
$$||Av|| = \sqrt{(\cos(\theta) + 2\sin(\theta))^2 + (3\cos(\theta) + 4\sin(\theta))^2}$$
Wolfram Alpha tells me that this expression is maximized by $\theta \approx -2.18468$, which results in $v \approx \begin{matrix}[-0.576 & -0.8174]^T\end{matrix}$, and for this $v$, we have $||Av|| \approx 5.465$. Thus $||A|| \approx 5.465$.

By the way, the way I calculated the norm above is not how you would do it in practice, especially for larger matrices. There is a key theorem which says that the 2-norm of a matrix $A$ is the largest singular value of $A$, which equals the square root of the largest eigenvalue of $A^* A$, where $A^*$ is the conjugate transpose of $A$. This fact is almost obvious if you know the geometrical meaning of the singular value decomposition. See Trefethen and Bau, Numerical Linear Algebra for a really nice discussion of this, and of matrix norms in general.

There are also short cuts for computing the 1-norm and $\infty$-norm. These are the maximum absolute column sum and maximum absolute row sum, respectively. Both of these short cuts are pretty easy to prove.

I found it useful to visualize what happens when a matrix transforms the vector. The attached images show a Mathcad implementation. The first image shows one way of calculating the norm and finding the angle for which the norm is a maximum (another might be to find where the derivative is zero). I used jbunniii's value for A to verify the solution. Note one difference (which should be slightly more obvious in the plot) is that I have chosen to look for the "maximum" angle within the range 0 to ∏, so I get a different value - however, as the norm function is cyclic, this doesn't really matter, the result is the same.

The second image plots the norm function versus angle, making clear its cyclic nature. It also plots the unit circle (v(θ)) and the circle as transformed under A (a(θ)). The right hand of the 2 lower plots connects the values for v to those of a, thus making the mapping explicitly visible. I changed the value of A to show the mapping more clearly (the original value of A results in a messier diagram!).

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ericm1234 said:
My book goes on to say:
"If we consider both C^n and C^m with norms, then we define the norm of an M x N matrix A by.."
Then the formula says norm of A=sup (over abs(v)=1) of abs(Av) = sup (over v does not equal 0) abs(Av)/abs(v)
Can someone please provide me at least one example of what this means?
If you read the LaTeX guide, you will be able to write things like this:
$$\|A\|=\sup_{|v|=1}|Av| =\sup_{v\neq 0}\frac{|Av|}{|v|}.$$ Hit the quote button to see how I did this.

Do you understand the concept of "supremum" (="least upper bound")?

The idea here is that the norm of A tells us how "long" A can make a unit vector. ##\|A\|## is the smallest real number such that no |Av| with |v|=1 is bigger. Equivalently, ##\|A\|## is the radius of the smallest closed ball around 0 that contains A(S), where S is the unit sphere.

Note that
$$\left\{|Av|:v\in\mathbb C^N, |v|=1\right\}=\left\{\frac{|Av|}{|v|}:v\in\mathbb C^N, v\neq 0\right\}.$$ Since the sets are the same, their supremums (least upper bounds) are the same.

This norm is more interested on infinite-dimensional vector spaces, because there the corresponding sets may not be bounded from above. If it is bounded from above, then the operator is said to be bounded. One of the theorems in functional analysis says that a linear operator is continuous if and only if it's bounded.

Actually, now that I think of it, that theorem is interesting here too, because it implies that every linear transformation between finite-dimensional vector spaces is continuous (with respect to the topology induced by the standard norm).

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## 1. What is a matrix norm?

A matrix norm is a mathematical concept that measures the "size" or magnitude of a matrix. It is analogous to the absolute value of a real number or the magnitude of a vector.

## 2. How is a matrix norm calculated?

The calculation of a matrix norm depends on the specific type of norm being used. Some common matrix norms include the Frobenius norm, spectral norm, and max norm. Each of these has a different formula for calculation.

## 3. What is the purpose of using a matrix norm?

The use of a matrix norm allows for the comparison of matrices based on their "size" or magnitude. This can be useful in various applications, such as optimization problems, linear algebra, and data analysis.

## 4. Can you provide an example for calculating a matrix norm?

Sure, let's take the matrix A = [1 2; 3 4]. The Frobenius norm of A can be calculated as: ||A||F = √(1² + 2² + 3² + 4²) = √30. The spectral norm of A can be calculated as: ||A||2 = 5. The max norm of A can be calculated as: ||A|| = max{|1|, |2|, |3|, |4|} = 4.

## 5. How is a matrix norm used in data analysis?

In data analysis, matrix norms can be used to measure the "size" or magnitude of a dataset, which can be useful in identifying patterns and relationships between variables. For example, the spectral norm can be used to identify the most important features in a dataset, while the Frobenius norm can be used to compare the overall magnitude of different datasets.

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