Eigenvalue and transfer function (2)

Scalar, vector, and function


When I want to mention a quantity, I use numbers. A number always represents ``something.'' For example, if I said 130, what is this number means? The number itself has not so much meaning. This could be someone is 130cm tall, or Autobahn's speed limit is 130km/hour.  One number represent ``something'' e.g., ``tall cm'', ``speed limit km/hour.''  Figure 1 shows this. These single numbers are Scalers. It is just a number, why it has a special name Scalar? I think this is just for distinguishing a scalar and a vector (or a (complex) number).

Figure 1 Scalar has some meaning


There are many stuffs I can not represent with a scalar. For instance, a place. The distance from my apartment to Zoo station can be represented by a scalar value. But, if you need a direction, I can't tell it by a scalar value. I could say, 50 degree from the north in clockwise direction, the distance is 5 km. Or go north 3 km and then go to 4km east. However, I can not tell you it by a single scalar value. In this case, a single scalar value is not enough and we should use many scalar values to express these entities. For example, if I want to record someone's ``tall'', ``weight'', ``current body temperature'', and ``blood pressure.'' Figure 2 shows such example.  The astute reader realized ``How can I put different kind of stuffs at the same axis?'' Here I showed hight and weight, they have apparently different units. (Although, in my school, we add the credit of math, gymnastics, and English. We did such an outrageous without any doubt. Therefore, one might not care about this, though, I think you should know this is a dangerous operation.) This is actually no problem if it is a vector. A vector is a sequence of scalars. For instance, we use two scalars to represent a position. They are an angle and a distance. These have also different unit. The problem occurs when you operate (plus, minus, multiply, dot product...) the vectors. Of course we can not add angle and distance. But, if we don't do such operation, these vectors still have some meaning.

Here I use [] to indicate this is a vector and put some number sequence. For example, [170 70 36 120] is a vector that has 4 scalars. The order of the scalars is important.

Figure 2 Vector is a series of scalar

Next I would like to talk about a relationship between vector and function. The relationship between scalar and vector seems well known. But the relationship between vector and function might not be known well. (I remember that I learned this in a Fourier transformation class.)


Eigenvalue and transfer function (1)


There was a famous mathematician and computer scientist, Richard Hamming. I am reading his book, Digital Filters. I would like to write  something I understand about this book.

Let's talk about eigenvalue and transfer function. But this is too sudden. Most of the people (including me) would say What is eigen-blah stuff? Therefore, I would like to start why it matters, what is the motivation to think about that, as usual in my blog. After reading some math book, I often said, ``I don't understand'' or ``So what?'' I want to say, ``Wow, that's great.'' If I said, ``Wow, that's great,'' then I usually understand what the purpose is and it is achieved in the paper. I try to explain this in the high school math only, but, I found out one step is missing. That is the relationship among scaler, vector, and function. I would like to explain these are all the same in some abstraction sense. Maybe high school students know a vector is a extension of a scalar, or a scaler is a special case of a vector. But, they might not know a function is also the extension of a vector. Let's start with such story.