How do I construct a matrix that satisfies certain conditions?

[Agent M27]

Homework Statement

I am having trouble building a matrix in my post so I have attached the question (2.27) and my work thus far. I need to develop a matrix H which when multiplied by a vector v produces a new matrix U with a constant 1/2 applied to certain terms of the matrix. Please see the attached matrix question for the final matrix which is desired. In the question v is composed of n elements, where n=8. From this I have been able to deduce that my final matrix will be a 4 x 1, my matrix for representing the vector v will be an 8 x 1 matrix, which forces my input matrix H to be a 4 x 8 matrix, due to the rules of matrix multiplication.

Homework Equations

The Attempt at a Solution

I realized that my matrix H must have two entries of 1/2 in each row. As far as the original question, my method proves correct, but when I look at the second part of the question, part (b), it alludes to the fact that my matrix H must be orthogonal, which in its current form it is not. I have attempted to place \frac{\sqrt{2}}{2} in clever locations along the rows which gives me orthogonality, but it does not fit the final condition of each row in matrix U being composed of only two elements of v and 1/2. Any help is greatly appreciated and sorry for the sloppy hand written solution. As an aside, can anyone let me know how to construct a matrix in this forum? Thanks in advance. BTW this problem is from the text "Discrete Wavelet Transformations: An Elementary Approach with Applications." by Patrick Van Fleet.

 

[gabbagabbahey]

I think that for this problem, it might be easiest if you work in index notation. Can you think of a way to represent H_{ij}?

Hint: The Kronecker delta should come in handy here

 

[Agent M27]

I also ought to have mentioned I have no experience or formal training what so ever concerning linear algebra, I am basically learning it from this text alone. This text is being used as a primer for a research project I am involved in. I briefly read up about the Kronecker Delta, but it is unclear to me how it would benefit me in solving this problem. Thanks.

Joe

 

[gabbagabbahey]

I also ought to have mentioned I have no experience or formal training what so ever concerning linear algebra, I am basically learning it from this text alone. This text is being used as a primer for a research project I am involved in. I briefly read up about the Kronecker Delta, but it is unclear to me how it would benefit me in solving this problem. Thanks.

Joe

Well, let's take a look at H for n=8:

H=\frac{1}{2}\begin{pmatrix}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{pmatrix}

Clearly, all the entries in a given row are zero except for two and those two are equal to 1/2 and depend on which row you are talking about. In the 1st row, only the 1st two entries are non-zero, so

H_{1j}=\left\{\begin{array}{lr}\frac{1}{2}, & j=1 \text{ or } 2 \\ 0, & j\neq 1 \text{ or } 2\end{array}\right. = \frac{1}{2}\left(\delta_{1,1}+\delta_{1,2}\right)

In the second row, only the 3rd and 4th entries are non-zero, so

H_{2j}=\left\{\begin{array}{lr}\frac{1}{2}, & j=3 \text{ or } 4 \\ 0, & j\neq 3 \text{ or } 4\end{array}\right. = \frac{1}{2}\left(\delta_{2,3}+\delta_{2,4}\right)

And so on... Can you deduce the general formula for H_{ij}?

 

[Agent M27]

I can only partially generalize the matrix H. I formed the following parameters:

H_ij = \frac{1}{2} when j=2i

Am I at least on the right track? Thanks for your patience, this linear algebra is a really brain burner!

By the way I tried to copy your latex code by clicking it and copying it from the window that it is brought up in, but it just comes out as the code, not the format as you have it, what did I do wrong?

Joe

 

[gabbagabbahey]

I can only partially generalize the matrix H. I formed the following parameters:

H_ij = \frac{1}{2} when j=2i

Am I at least on the right track? Thanks for your patience, this linear algebra is a really brain burner!

Close, H_{ij} also equals 1/2 when j=2i-1. So, the general expression will be

H_{ij}=\frac{1}{2}\left(\delta_{2i,j}+\delta_{2i-1,j}\right)

Follow?

By the way I tried to copy your latex code by clicking it and copying it from the window that it is brought up in, but it just comes out as the code, not the format as you have it, what did I do wrong?

Joe

Just enclose the code in tex tags ([#tex] put code here and get rid of the # signs [#/tex])