# Can someone please check this work (eigenvalues)

• iamsmooth
In summary, the conversation is about finding an invertible S and a diagonal D that satisfies the equation S^-1AS=D. The student has found a solution with S and D, but their professor has pointed out that the eigenvalues in D are incorrect. After some discussion, the student realizes that they forgot to divide by the determinant when finding the inverse of S. They then correct their solution and confirm that it is now correct.
iamsmooth

## Homework Statement

Let

$$A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right]$$

Find an invertible S and a diagonal D such that $$S^{-1}AS=D$$

## Homework Equations

I basically have the question answered, just ONE problem.

## The Attempt at a Solution

$$S = \left[ \begin{array}{cc} 1 & -1 \\ 14 & 2 \end{array} \right]$$

$$D = \left[ \begin{array}{cc} -40 & 0 \\ 0 & -104 \end{array} \right]$$

I asked the professor what was wrong and he said "check the eigenvalues (main diagonal of D). The eigenvectors look ok, but their order has to match that of the eigenvalues.

I've checked over and over, the math works out(S^-1 A S = D) , but I can't see what's wrong with the order. It looks perfectly fine for me.

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The math does not work out 'fine'. S^(-1)AS=[[-5/2,0],[0,-13/2]] as I work it out. I'll agree with your professor. The eigenvalues are wrong, but the eigenvectors are right. How did you do that?

Well D is what I got from multiplying inverse of S, which was the eigen vectors I got multiplied by A, and then again by S, so...

$$\left[ \begin{array}{cc} 2 & 1 \\ -14 & 1 \end{array} \right] \left[ \begin{array}{cc} -6 & .25 \\ 7 & -3 \end{array} \right] \left[ \begin{array}{cc} 1 & -1 \\ 14 & 2 \end{array} \right]$$

Which gave me my D which I guess is obviously wrong. $$D = \left[ \begin{array}{cc} -40 & 0 \\ 0 & -104 \end{array} \right]$$

My eigenvalues I got were 6.5 and 2.5 from the characteristic polynomial. Is that what's supposed to go into D's main diagonal or something? I thought I had to multiply it out like that.

That helps. Your inverse of S is wrong. Try multiplying (inverse S)*S. You don't get the identity. Do you? Did you forget to divide by a determinant? And yes, the diagonal of D should be the eigenvalues.

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Oh dear, I did forget about that! Thanks! I'll try again.

Okay so,

$$S = \left[ \begin{array}{cc} 1 & -1 \\ 14 & 2 \end{array} \right]$$

$$S = \left[ \begin{array}{cc} -2.5 & 0 \\ 0 & -6.5 \end{array} \right]$$

Is this perfectly correct? I only get one more submission for this, so I need to make sure this is 100% correct. I'm only unsure because the prof said that "the eigenvectors look okay, but order has to match eigenvalues). Is that matched?

You mean D= on the second line, right? And there's more than one S that will diagonalize A. So I'm not promising your homework system will accept it. But that looks ok to me. At least you have the eigenvalues on the diagonal.

Whoops yeah, I meant D, and yeah I found that there were other values of S that would diagonalize A, some of which were just switching the signs of the elements around. I think you're right, the answer should be correct now.

Well, you'll have to trust that the homework scanner will accept any correct answer. (You could also multiply S by 2 and multiply (inverse S) by 1/2.) If not, you have good grounds for appeal.

Yeah the answer is correct now. All this new material made me forget the earlier stuff :(

Thanks a lot, appreciate the help.

## What are eigenvalues and why are they important in mathematics?

Eigenvalues are a concept in linear algebra that are used to understand the behavior of linear transformations and matrices. They represent the scaling factor by which a vector is stretched or compressed when multiplied by a matrix. Eigenvalues are important because they can help us solve systems of linear equations, identify important patterns in data, and analyze the stability of dynamic systems.

## How do you find eigenvalues?

To find eigenvalues, we first need to find the characteristic polynomial of the matrix. This is done by subtracting the identity matrix multiplied by the variable λ from the given matrix and then finding the determinant. The resulting polynomial will have λ as its variable. The roots of this polynomial are the eigenvalues.

## What is the relationship between eigenvalues and eigenvectors?

Eigenvectors are the vectors that are associated with specific eigenvalues. When a matrix is multiplied by an eigenvector, the result is a new vector that is parallel to the original eigenvector. The eigenvalue is the factor by which the eigenvector is scaled. In other words, the eigenvector tells us the direction in which the matrix is stretched or compressed, and the eigenvalue tells us the magnitude of this transformation.

## Can you have multiple eigenvalues for a single matrix?

Yes, it is possible for a matrix to have multiple eigenvalues. This means that there are multiple scaling factors that can be associated with different eigenvectors for the same matrix. In fact, the number of eigenvalues for a matrix is equal to its dimension. However, some matrices may have repeated eigenvalues, which means that there are fewer distinct eigenvalues than the dimension of the matrix.

## How are eigenvalues used in real-world applications?

Eigenvalues have many practical applications in various fields such as physics, engineering, economics, and computer science. They can be used to analyze complex systems and predict their behavior, solve differential equations, compress data, and perform image processing. For example, in quantum mechanics, the eigenvalues of a particle's energy operator represent the possible energy states of the particle. In Google's PageRank algorithm, the eigenvalues of a matrix are used to rank web pages based on their importance.

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