# Finding eigenvalues and eigenspaces with only this info

• Dist
In summary, there is no way to find the other eigenvalues and eigenvectors of a 3x3 matrix with one known eigenvalue and eigenvector, whether they correspond or not. This is because there are an infinite number of matrices that could have the given eigenvalue and eigenvector, making it impossible to determine the other eigenvalues and eigenvectors without more information.
Dist
Lets say I have a 3x3 matrix 'A' and one known eigenvalue 'z' and one known eigenvector 'x', but they don't "belong" to each other, as in Ax =/= zx
Is there a way of finding the other eigenvalues and eigenspaces of A using only this piece of information?
Thanks.

I doubt it.

Just count the amount of information needed and available.
A 3x3 matrix has 9 numbers.
An eigenvalue is one number.
An eigenvector is two numbers (it's only defined upto a scale factor)
Thus in total you have 3<9 pieces of information.

Note that the full eigensystem is 3 eigenvalues and 3 eigenvectors
= 3*1 + 3*2 = 3*(1+2) = 9 pieces of information.

Even if the eigenvalue and eigenvector "belong" to each other, the other eigenvalues and eigenvectors could be anything. There will exist an infinite number of matrices having the given eigenvector and eigenvalue.

Or are you saying that you have a known matrix, one eigenvalue and eigenvector, and want to know if there is a simpler way of finding the other eigenvalues and eigenvectors than the usual "solve the characteristic equation"?

No, the matrix A is unknown as well. And I only want to find the eigenvalues and eigenspaces of A, not necessarily its eigenvectors.

If you know the eigenspaces then you can choose a set of eigenvectors. So, your last statement is a little meaningless.

As I said before, given one eigenvalue and one eigenvector, whether they correspond or not, there exist an infinite number of matrices with those given eigenvalue and eigenvector and whatever other eigenvalues and eigenvectors you want to assign.

A single eigenvector and eigenvalue tell you nothing about other eigenvalues and eigenvectors.

Ok, I see, thanks.
But say we could choose arbitrary eigenvalues and eigenvectors that would go with the ones we already know. How would one go about doing that?

## 1. What are eigenvalues and eigenspaces?

Eigenvalues and eigenspaces are mathematical concepts used in linear algebra to describe the properties of a square matrix. Eigenvalues are numbers that represent the scaling factor of a particular eigenvector, while eigenspaces are the set of all vectors that satisfy a specific eigenvalue-eigenvector equation.

## 2. How can I find eigenvalues and eigenspaces?

To find eigenvalues and eigenspaces, you need to first determine the characteristic polynomial of the given matrix. Then, you can use various methods such as the characteristic equation or the diagonalization method to solve for the eigenvalues and corresponding eigenspaces.

## 3. Can I find eigenvalues and eigenspaces with limited information?

Yes, it is possible to find eigenvalues and eigenspaces with only limited information. However, the more information you have about the matrix, the easier and more accurate the calculation will be.

## 4. What are the applications of finding eigenvalues and eigenspaces?

Finding eigenvalues and eigenspaces is commonly used in various fields such as physics, engineering, computer science, and economics. It is used to solve systems of differential equations, analyze the stability of dynamic systems, and perform data compression and dimensionality reduction.

## 5. Are there any software or tools to help with finding eigenvalues and eigenspaces?

Yes, there are many software and tools available that can help with finding eigenvalues and eigenspaces. Some popular ones include MATLAB, Mathematica, and Python libraries such as NumPy and SciPy. These tools use efficient algorithms to quickly and accurately calculate eigenvalues and eigenspaces for large and complex matrices.

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