How to find first matrix of SVD?

  • Thread starter zak100
  • Start date
  • #1
462
11

Homework Statement


I dont know how to find the first matrix of SVD. I know how to find the middle one and the last one. For first one some tutorials found AV1. I dont know how to find it. Is there any simple way to find the first matrix.
2. Homework Equations [/B]
SVD = A*Summation matrix * V^t

The Attempt at a Solution



A * A^t= \begin{bmatrix}0&1&1\\1&0&1\\end{bmatrix}\cdot \begin{bmatrix}0&1\\1&0\\1&1\end{bmatrix}

= \begin{bmatrix}2&1\\1&2\\end{bmatrix}

========
If possible please correct my Latex also


I have got the following matrix. I have found the eigen values

A*A^t__=__[0__1__1]*__________________________[0____1]
___________[1__1__0]___________________________[1____0]
_______________________________________________[1____1]
______________________________________________________
=[2____1]______________________________________________
_[1____2]______________________________________________


The Attempt at a Solution





λ1=1 & λ2=3, σ1=sqrt(3) & σ2=1

υ1 =1/sqrt(2)[1]____________________________________________
_____________[1]____________________________________________
& υ2= 1/sqrt(2)[1]_____________________________________________
________________[-1]____________________________________________
Some body please guide me how to find the first matrix of SVD.

Zulfi.
 

Answers and Replies

  • #2
StoneTemplePython
Science Advisor
Gold Member
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Homework Statement


I dont know how to find the first matrix of SVD. I know how to find the middle one and the last one. For first one some tutorials found AV1. I dont know how to find it. Is there any simple way to find the first matrix.
2. Homework Equations [/B]
SVD = A*Summation matrix * V^t

The Attempt at a Solution



A * A^t= \begin{bmatrix}0&1&1\\1&0&1\\end{bmatrix}\cdot \begin{bmatrix}0&1\\1&0\\1&1\end{bmatrix}
I presume we're dealing with real scalars here. you want the SVD of A. Solution: Transpose A, re-rerun your process to find "the last one" and call it ##\mathbf U^T##, transpose again, and now you've found ##\mathbf U##. In both cases make sure you've ordered the singular values in the same way : I'd suggest: ##\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n \geq 0##. Problem solved.
- - -
Note: I've seen a repeating theme in your threads: your "relevant equations" are consistently wrong. This is deeply problematic. You need correct simple things, called relevant equations, that you can build a correct answer off of.

In this thread, you want the SVD of A, as implied by the steps in your attempt at a solution. So the SVD is given by

##\mathbf A = \mathbf U \mathbf \Sigma \mathbf V^T \neq \mathbf A \mathbf \Sigma \mathbf V^T =\text{what your relevant equation says}##

where I'm being a good sport and allowing ##\mathbf \Sigma = \text{Summation matrix}##
 
  • #3
462
11
Hi,

Thanks for your response and for correcting my equation. I cant understand what you mean by "the last one" "

If possible please give an example. This looks a different method from what i am seeing. They are doing it by finding rank of a matrix.

When the teacher taught, i thought \mathbf \Sigma = \text{Summation matrix}
but then he gave an example, it got clear that this is something else. Any way its my wrong belief that I still use Sigma symbol for summation.

Zulfi.
 
  • #4
StoneTemplePython
Science Advisor
Gold Member
1,180
581
Hi,

Thanks for your response and for correcting my equation. I cant understand what you mean by "the last one" "

If possible please give an example. This looks a different method from what i am seeing. They are doing it by finding rank of a matrix.

When the teacher taught, i thought \mathbf \Sigma = \text{Summation matrix}
but then he gave an example, it got clear that this is something else. Any way its my wrong belief that I still use Sigma symbol for summation.

Zulfi.
please but double hash tags on both sides of your math latex to make it render.

I put "the last one" in quote because I am quoting you here.

I know how to find the middle one and the last one.
So, no I don't believe you cannot understand what I mean by "the last one" -- I am referring precisely to the two matrices you referenced in your original post from 43 minutes ago.
 
  • #5
462
11
Hi,
Lets suppose, this one is my
A=[0__1__1]
__[1__1__0]

You mean following stuff

A^T*A=[0____1]*______________[0____1_____1]_________________________
______[1____1]_______________[1____1_____0]_________________________
______[1____0]____________________________________________________
_________________________________________________________________
=[0+1_______0+1_______________0+0]________________________________
_[0+1_______1+1_______________1+0]________________________________
_[0+0_______1+0_______________1+0]_________________________________

=[1_________1_________________0]__________________________________
_[1_________2_________________1]__________________________________
_[0_________1_________________1]__________________________________

Please guide me if its correct or not.

Zulfi.
 
  • #6
StoneTemplePython
Science Advisor
Gold Member
1,180
581
Hi,
Lets suppose, this one is my
A=[0__1__1]
__[1__1__0]

You mean following stuff

A^T*A=[0____1]*______________[0____1_____1]_________________________
______[1____1]_______________[1____1_____0]_________________________
______[1____0]____________________________________________________
_________________________________________________________________
=[0+1_______0+1_______________0+0]________________________________
_[0+1_______1+1_______________1+0]________________________________
_[0+0_______1+0_______________1+0]_________________________________

=[1_________1_________________0]__________________________________
_[1_________2_________________1]__________________________________
_[0_________1_________________1]__________________________________

Please guide me if its correct or not.

Zulfi.
Look, I'm not sure what you're doing here. I'll try quoting you one last time.

Your post is titled "How to find first matrix of SVD", and
I dont know how to find the first matrix of SVD. I know how to find the middle one and the last one... Is there any simple way to find the first matrix.
and as I said in post #2 the answer is yes. And it is very simple.

you have Singular Value Decomposition of a matrix ##\mathbf A## given by

##\mathbf A = \mathbf U_A \mathbf \Sigma_A \mathbf V_A^T ##

and the SVD of some matrix ##\mathbf B## given by

##\mathbf B = \mathbf U_B \mathbf \Sigma_B \mathbf V_B^T ##

(a small point: in both cases I suggested the singular values should be ordered from biggest to smallest from top left to bottom right of the middle matrices, which are diagonal)

You have told us that you already know how to find the middle and last matrix, which is given by ##\mathbf \Sigma_A \mathbf V_A^T## in the first case, and by ##\mathbf \Sigma_B \mathbf V_B^T## in the second case. But in particular, you are interested in finding ##\mathbf U_A## and don't know how to find it.

My point is assign ##\mathbf B:= \mathbf A^T##.

The insight you need is to recognize

##\mathbf U_A^T = \mathbf V_B^T##
so to find the matrix you want, transpose it and get

##\text{matrix you want to find} = \mathbf U_A = \mathbf V_B=\text{matrix you already know how to find}##

Problem solved.
- - - -
Equivalently, this thread doesn't make much sense to me because if you already "know how to find the middle one and the last one", then you actually already know how to find "the first matrix". I am pointing that out to you now, for some analytical insights and also to save you a ton of work.

If you now tell me now that you don't "know how to find the middle one and the last one", then that's going to be a problem.
 
  • #7
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
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1,031
I am editing the OP by fixing some of the LaTeX code. You definitely left out the ## \text{ ## } ## at the beginning and at the end of each expression.
It's good to see you attempt to use LaTeX.

Homework Statement


I dont know how to find the first matrix of SVD. I know how to find the middle one and the last one. For first one some tutorials found AV1. I dont know how to find it. Is there any simple way to find the first matrix.
2. Homework Equations [/B]
SVD = A*Summation matrix * V^t

The Attempt at a Solution



A * A^t= \begin{bmatrix}0&1&1\\1&0&1\\end{bmatrix}\cdot \begin{bmatrix}0&1\\1&0\\1&1\end{bmatrix}
## A A^T= \begin{bmatrix}0&1&1\\1&0&1\end{bmatrix}\cdot \begin{bmatrix}0&1\\1&0\\1&1\end{bmatrix} ##

## = \begin{bmatrix}2&1\\1&2\end{bmatrix} ##​
========
If possible please correct my Latex also

I have got the following matrix. I have found the eigen values

A*A^t__=__[0__1__1]*__________________________[0____1]
___________[1__1__0]___________________________[1____0]
_______________________________________________[1____1]
______________________________________________________
=[2____1]______________________________________________
_[1____2]______________________________________________

The Attempt at a Solution



λ1=1 & λ2=3, σ1=sqrt(3) & σ2=1
##\lambda_1=1\,,\ \lambda_2=3\,,\ \sigma_1=\sqrt{3} \,,\ \sigma_2=1 \ ##
υ1 =1/sqrt(2)[1]____________________________________________
_____________[1]____________________________________________
& υ2= 1/sqrt(2)[1]_____________________________________________
________________[-1]____________________________________________
I'm not quite sure what you mean by the above. Maybe a attempt at matrix U .
Some body please guide me how to find the first matrix of SVD.

Zulfi.
 
  • #8
462
11
Hi,

Thanks SammyS, Good work. When i do right click, I cant see double hash with the math stuff.

Thanks StoneTemplePython. Actually I cant understand your algorithm & this was simple way to achieve my work i.e Solution: Transpose A, re-rerun your process to find "the last one" . Again the process was related to finding the eigen values and eigen vectors which i overlooked.


##\mathbf U, V ## both are vector matrices formed using eigen vectors.


##\mathbf U= A * A^T ## &

##\mathbf V= A^T * A##


Zulfi.
 
  • #9
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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1,728
Hi,
Lets suppose, this one is my
A=[0__1__1]
__[1__1__0]

You mean following stuff

A^T*A=[0____1]*______________[0____1_____1]_________________________
______[1____1]_______________[1____1_____0]_________________________
______[1____0]____________________________________________________
_________________________________________________________________
=[0+1_______0+1_______________0+0]________________________________
_[0+1_______1+1_______________1+0]________________________________
_[0+0_______1+0_______________1+0]_________________________________

=[1_________1_________________0]__________________________________
_[1_________2_________________1]__________________________________
_[0_________1_________________1]__________________________________

Please guide me if its correct or not.

Zulfi.
Nobody understands what you are doing here, so you might as well not even bother writing it. Use an understandable notation, and take the time to explain what you are attempting to calculate, using perhaps a few words as well as symbols. Above all: if you are entering tables or matrices, use LaTeX. You can actually construct fairly nice-looking tables in the version of LaTeX available here, just by using an array. For example:
$$ \begin{array}{rccl}
\text{Type} & A & B & \text{Total}\\
1 & 100 & \alpha & 100+\alpha\\
2 & 50 & 150 & 200\\
\text{Total}& 150 & \alpha +150 & 300 + \alpha
\end{array}
$$
You can even fix it up using horizontal and vertical dividers:
$$ \begin{array}{r|cc|l}
\text{Type} & A & B & \text{Total} \\ \hline
1 & 100 & \alpha & 100+\alpha\\
2 & 50 & 150 & 200\\ \hline
\text{Total}& 150 & \alpha +150 & 300 + \alpha \\

\end{array}
$$
To see how it is done, just right-click on the images and ask for a display of math as tex commands.
 
Last edited:
  • #10
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,359
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Hi,

Thanks SammyS, Good work. When i do right click, I cant see double hash with the math stuff.

Zulfi.
Zulfi,
Using the 'right click' only works on LaTeX images you find in finished posts, i.e. posts which have been submitted.

To see is your LaTeX code is going to work properly, use the "PREVIEW..." tab, found in the lower right corner as you are composing your Post.
This is mentioned in the 4th paragraph of the introduction to the PF LaTeX Primer, and repeated below.
To practice your LaTeX, enter it in the editing box for a new thread or post, and click the "Preview" button. Don't click the "Create Thread" or "Post Reply" button unless you want everybody else to see what you're doing!​
.
 
Last edited:
  • #11
462
11
Thanks. I would try as much as possible.

Zulfi.
 

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