Yes I assumed they were of the same dimension, otherwise it wouldn't be invertible, but I have a problem with your statement. Consider for instance the projection
T(x,y) = (x,0)
T receives a bidimensional vector and also returns one, soT:R^{2}→R^{2}. The spaces are of the same dimension, but...
I was struck with the following question: Is there a linear map that's injective, but not surjective? I know full well the difference between the concepts, but I'll explain why I have this question.
Given two finite spaces V and W and a transformation T: V→W represented by a matrix \textbf{A}...
First of all, seeing we're obviously discussing discrete signals let's make a few things clear. The Fourier Transform of a 1D signal can be defined over \mathbb{R}, unlike the Discrete Fourier Transform which results in a discrete function. On the other hand, the Z-Transform is a function...
These problems are rarely straightforward.
First of all, you shouldn't solve a problem in a certain way just because it's how it's usually done. That usually means you don't understand what you're doing. If there is a solution to this optimization problem then there must be a set of...
I see your point, but you still need to define a norm or else the space isn't metric, so saying it an euclidean space is by definition \mathbb{R}^n still isn't correct. However, based on what you said, if you define a norm in \mathbb{R}^n then the space is instantly euclidean, regardless of the...
Feel free to correct me, but saying Euclidean space is by definition \mathbb{R}^n may be a bit of an abuse.
\mathbb{R}^n is a vector space generated by the span of n linearly independent vectors also with dimension n. This space, however, needn't be Euclidean as on its own it lacks an inner...
OK I solved this, so I might as well share it with everyone else.
First of all, for convenience, I changed the constraint to R RT=I, which is completely equivalent.
So the problem is as follows:
Maximize tr{R_{xy}R^{T}} subject to R R^{T}=I
The solution:
First of all, we can...
I'll try this tomorrow, but it will most likely give the desired result. However, why are we given such freedom of defining a new matrix product that's more suitable to the problem? Or is it that in these types of constraints this alternative product must be used?
Do you know of any...
The problem:
Minimize tr{RyxR} subject to RTR=I
This problem is known as Procruses Analysis and can be solved using Lagrange Multipliers, so there's a tendency to write the following function:
L(R) = tr{RyxR} - \Lambda(RTR-I),
where \Lambda is a matrix of Lagrange Multipliers
However, there...