Solving Zero Eigenvector: A Homework Problem

In summary, the conversation discusses solving for eigenvectors in situations where a zero vector appears to be the only solution. Maple gives specific eigenvalues and eigenvectors for a given matrix, but the attempt at a solution by hand does not yield the same results. The solution is found by realizing that the two equations are equivalent, and any vector that satisfies either equation can be used as the eigenvector. This type of situation is called a redundancy.
  • #1
Ryan007
14
1

Homework Statement



I can calculate the proper eigenvalues, but when I plug them back into the matrix, I get x1=0 and x2=0. But this is not the answer Maple gives me! How do I solve for the eigenvector when it appears that a zero vector is the only solution?

Homework Equations



For example, for the matrix {1,1},{1,-1} (rows shown), Maple gives me: eigenvalue of sqrt(2) with eigenvector {1/((sqrt(2)-1),1} and eigenvalue of -sqrt(2) with eigenvector {1/(-(sqrt(2)-1),1}

The Attempt at a Solution



But I can't get these eigenvectors when I try to solve by hand! How do you solve in these situations?
What are these situations called?



 
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  • #2
For sqrt(2), You end up with the eigenvector equation:

[tex]\left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)\binom{x}{y}=\sqrt{2}\binom{x}{y}[/tex]

or:

[tex](1-\sqrt{2}) x+y=0[/tex]

[tex]x-y(1+\sqrt{2})=0[/tex]

which are redundant so any vector (x,y) that satisfies either equation is a suitable eigenvector like:

[tex]\binom{1+\sqrt{2}}{1}[/tex]

and even:

[tex]\binom{\frac{1}{\sqrt{2}-1}}{1}[/tex]
 
  • #3
Thanks! Oh duh! I didn't realize that the 2 equations were the same. That's why I was getting x=0 and y=0 as solutions. I had to multiply through by the appropriate constant to make the 2 equations look the same.
 

What is a zero eigenvector?

A zero eigenvector is a vector that, when multiplied by a matrix, results in a vector of all zeros. This means that the vector does not change direction or magnitude when multiplied by the matrix.

Why is solving for a zero eigenvector important?

Solving for a zero eigenvector is important because it can provide valuable information about the matrix. It can indicate if the matrix has a nontrivial solution, which can be useful in various applications such as solving systems of linear equations.

What is the process for solving a zero eigenvector?

The process for solving a zero eigenvector involves finding the null space, or kernel, of the matrix. This can be done by setting up a system of linear equations and solving for the variables that result in a vector of all zeros. Alternatively, the null space can be found using matrix operations such as row reduction.

What are some real-life applications of solving for a zero eigenvector?

Solving for a zero eigenvector has various real-life applications, including image processing, signal processing, and circuit analysis. It is also commonly used in data analysis and machine learning algorithms.

Can there be more than one zero eigenvector for a matrix?

Yes, there can be multiple zero eigenvectors for a matrix. This is because a matrix can have multiple linearly independent vectors that result in a vector of all zeros when multiplied by the matrix. These vectors are known as the eigenvectors of the matrix.

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