Find a 2x2 Matrix A for Given Eigenspaces E_2 and E_4

In summary, to find a 2x2 matrix A where E_4 = span [1,-1] and E_2 = span [-5, 6], you can construct the matrices C and D directly by solving the system of equations formed by multiplying A with the given eigenvectors and corresponding eigenvalues.
  • #1
snoggerT
186
0
Find a 2\times 2 matrix A for which

E_4 = span [1,-1] and E_2 = span [-5, 6]

where E_(lambda) is the eigenspace associated with the eigenvalue (lambda)


relevant equations: Av=(lambda)v

The Attempt at a Solution



I've pretty much gotten most of the eigenspace/value problems down, but this one I'm clueless on how to work. Seems that you would have to work backwards , but I don't know how to do that.
 
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  • #2
Suppose you have a matrix A and it is diagonalizable. That means you can write it as
[tex]A = C D C^{-1}[/tex]
When A is given, how can you construct the matrices C and D?

Once you answer this question, you'll also see the answer to your problem.
 
  • #3
You can just do it directly. You are told that
[tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}1 \\ -1\end{array}\right)= 4\left(\begin{array}{c}1\\ -1\end{array}\right)= \left(\begin{array}{c}4\\ -4\end{array}\right)[/tex]
which gives you two equations for a, b, c, d and
[tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}-5 \\ 6\end{array}\right)= 2\left(\begin{array}{c}-5\\ 6\end{array}\right)= \left(\begin{array}{c}-10\\ 12\end{array}\right)[/tex]
which gives you another two equations. Solve those 4 equations.
 

1. What is an eigenvalue problem?

An eigenvalue problem is a mathematical problem that involves finding the eigenvalues and corresponding eigenvectors of a given matrix or linear operator. Eigenvalues and eigenvectors are important concepts in linear algebra and are used to understand the behavior of linear systems.

2. How is an eigenvalue problem solved?

The most common method for solving an eigenvalue problem is through diagonalization, which involves finding a matrix that is similar to the original matrix and has the eigenvalues on its diagonal. Other methods include using the characteristic polynomial, power iteration, and QR algorithm.

3. What is the importance of eigenvalues in an eigenvalue problem?

Eigenvalues are important because they represent the scaling factor of the corresponding eigenvector. They are also used to understand the behavior of linear systems, such as stability and oscillations. In addition, eigenvalues are used in various applications such as image processing, data compression, and quantum mechanics.

4. Can an eigenvalue problem have more than one solution?

Yes, an eigenvalue problem can have multiple solutions. This is because a matrix can have repeated eigenvalues, and therefore, multiple eigenvectors can correspond to the same eigenvalue. In addition, some matrices may have complex eigenvalues, which means that the eigenvectors can also be complex.

5. How are eigenvalues and eigenvectors used in data analysis?

Eigenvalues and eigenvectors are used in data analysis to reduce the dimensionality of a dataset and extract the most important features. This is done through techniques such as Principal Component Analysis (PCA), where the eigenvectors of the covariance matrix represent the principal components of the data. Eigenvalues are also used to calculate the variance explained by each principal component.

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