Pre-quantum matrix problem class problem? Probably simple?

It is customary to order them in descending order of eigenvalue, so that the first column of U corresponds to the largest eigenvalue, and so forth.
  • #1
jeebs
325
4
Here is my problem. Consider the matrices

[tex]A = \left(\begin{array}{ccc}0&i&2\\0&1&0\\1&0&0\end{array}\right) [/tex]and [tex]B = \left(\begin{array}{ccc}2&i&0\\3&1&5\\0&-i&-2\end{array}\right)[/tex]

Calculate A-1B andBA-1. Are they equal?
Hint: find the inverse of A in the diagonal basis.

I am really not sure how to proceed with this. The "diagonal basis" hint made me think I should be looking to rewrite A like
[tex]A = \left(\begin{array}{ccc}A_1_1&0&0\\0&A_2_2&0\\0&0&A_3_3\end{array}\right) [/tex]

I was thinking I need eigenvalues to put on the leading diagonal so I attempted to use an eigenvalue equation [tex]A|u> = \lambda|u> [/tex], where I start with [tex] |u> = \left(\begin{array}{c}u_1\\u_2\\u_3\end{array}\right)[/tex].

If I do this I end up with the system of simultaneous equations:
[tex]iu_2 + 2u_3 = \lambda u_1[/tex]
[tex]u_2 = \lambda u_2[/tex]
[tex]u_1 = \lambda u_3[/tex]

From the 2nd one I can say that [tex]\lambda = 1[/tex] which gives me:
[tex]u_1 = u_3[/tex] and so [tex]|u> = \left(\begin{array}{c}u_1\\-u_1/i\\u_1\end{array}\right) = \left(\begin{array}{c}1\\-1/i\\1\end{array}\right) = \left(\begin{array}{c}1\\i\\1\end{array}\right) [/tex] which is normalized.

This problem was part of a past problem class I went to, and on the board was written "Diagonalize matrix: AD = U+AU" (where the + is meant to be a dagger really, denoting an adjoint). Looking back now this expression means nothing to me, but also we were told that apparently we should end up with [tex]U = \left(\begin{array}{ccc}-\sqrt{2}&sqrt{2}&1\\0&0&i\\1&1&1\end{array}\right) [/tex] and [tex]A_D = \left(\begin{array}{ccc}-\sqrt{2}&0&0\\0&\sqrt{2}&0\\0&0&1\end{array}\right) [/tex] and that the columns of U are eigenvectors. I do not know what the hell to make of this, but I do notice that the eigenvector that I worked out in the stuff I have just shown above is the same as the third column, so I must be at least somewhere near the right track. I'm sorry I cannot explain more but this is everything I have to go on from what I wrote down.

However, I do not know:
a) how I am supposed to come up with 2 other eigenvectors
b) why, once I have 3 eigenvectors, I should arrange them into this 3x3 object called U

I get the feeling this is actually a straightforward task to do, I'm just not clear on the way to do it. Can anyone shed any light on this? It would be greatly appreciated. Thanks.
 
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  • #2
As you are starting to realize, the matrix U that diagonalizes A has the eigenvectors of A as its columns. The problem you're having is that to set up the eigenvalue equation

[tex]A | u \rangle = \lambda | u\rangle,[/tex]

You must first determine the eigenvalues of A from the characteristic equation you get by rewriting this as

[tex](A -\lambda I)| u \rangle = 0, \rightarrow \det (A -\lambda I) =0.[/tex]

There will be 3 eigenvalues, finding the corresponding eigenvectors is an independent problem for each eigenvalue.

You've already found an eigenvector for [tex]\lambda =1[/tex]. You should verify that [tex]\lambda =1[/tex] is actually an eigenvalue of A.
 
  • #3
jeebs said:
If I do this I end up with the system of simultaneous equations:
[tex]iu_2 + 2u_3 = \lambda u_1[/tex]
[tex]u_2 = \lambda u_2[/tex]
[tex]u_1 = \lambda u_3[/tex]

From the 2nd one I can say that [tex]\lambda = 1[/tex]
Keep in mind you assumed u2≠0 to conclude λ=1. If u2=0, λ could be something else.
However, I do not know:
a) how I am supposed to come up with 2 other eigenvectors
The other two correspond to solutions where u2=0. You could proceed as you did, but the method fzero suggested is more straightforward.
 
  • #4
right, what I got for fzero's way was:

[tex]A - \lambda I = \left(\begin{array}{ccc}-\lambda&i&2\\0&1-\lambda&0\\1&0&-\lambda\end{array}\right) [/tex]

and the determinant of that I got was [tex] 2\lambda + \lambda^2 - \lambda^3 - 2 = 0 [/tex].

for this I get that [tex] \lambda = 1, +/-\sqrt{2} [/tex]. I'm not sure what to do with these now though, I mean, I see that they are the same as the AD diagonal matrix, but why would I then make the step of putting these in a matrix, and in what is to say in which order on the diagonal they go?
 
Last edited:
  • #5
Well first of all, find the other eigenvectors. You did this for [tex]\lambda=1[/tex] already, for the other eigenvalues, you can use the same equations, but set [tex]\lambda=\pm \sqrt{2}[/tex]. As for the matrix U, it really doesn't matter which eigenvector you use for which column, you'll just get an AD that differs by a permutation of eigenvalues on the diagonal.
 

Related to Pre-quantum matrix problem class problem? Probably simple?

1. What is a pre-quantum matrix problem class problem?

A pre-quantum matrix problem class problem refers to a mathematical problem that deals with the manipulation and analysis of matrices, which are arrays of numbers or symbols arranged in rows and columns. The term "pre-quantum" indicates that these problems are solved using traditional mathematical methods, rather than quantum computing techniques.

2. How is a pre-quantum matrix problem different from a regular matrix problem?

A pre-quantum matrix problem is typically more complex and challenging to solve compared to a regular matrix problem. This is because pre-quantum matrix problems often involve larger matrices and require advanced mathematical techniques to find a solution.

3. What is the significance of studying pre-quantum matrix problems?

Pre-quantum matrix problems have practical applications in various fields, including physics, engineering, and computer science. By studying these problems, scientists can better understand the behavior of complex systems and develop more efficient algorithms for solving them.

4. What are some common techniques used to solve pre-quantum matrix problems?

Some common techniques for solving pre-quantum matrix problems include Gaussian elimination, LU decomposition, and singular value decomposition. These methods involve manipulating the matrix to reduce it to a simpler form and then using algebraic operations to find a solution.

5. Are there real-world problems that can be modeled as pre-quantum matrix problems?

Yes, many real-world problems can be modeled and solved using pre-quantum matrix problems. For example, optimization problems in logistics and operations research, image and signal processing, and data analysis can all be represented as pre-quantum matrix problems.

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