How to find eigenvalues and eigenvectors for 5x5 matrix

In summary, if you are given exercises to find eigenvalues of 5 by 5 matrices by hand, it will likely involve using tricks or numerical approximations due to the tedious and error-prone nature of the process. Options for simplifying the process include using software such as MATLAB, Mathematica, Maple, or Wolfram Alpha. If you do not have access to these tools, you may have to manually find the nullspace through inspection or using methods like Newton's method.
  • #1
DavidLiew
16
0
I got a 5x5 matrix, if use characteristic equation to find the eigenvalues and eigenvectors are very tedious and trouble, so got any method which are easy to calculate?
 
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  • #2
unless you've got a nice matrix (read diagonal) you're going to have to use some tricks, if you're lucky you can use the tschirnhaus transformation but most likely you'll have to resort to numerical approximations for the eigenvalues (Newtons method or something) then you'll have to churn through to find the nullspace manually

it's pretty tedious work and chances are that you'd end up making an error doing it anyway..
you could just use mathematica or matlab, that'd be easier

if you don't have any of those, post your matrix and I'll give you the results, if you want?
 
  • #3
I need to do it in paper, can not use software to solve.
 
  • #4
in that case I'd say you're going to be stuck with inspection, Newtons method or something like it or if you're lucky you may be able to reduce it to an easier polynomial via some transformation

sorry bro
 
  • #5
If you are given exercises in which you are asked to find eigenvalues of 5 by 5 matrices by hand, I suspect it will have been set up so the characteristic equations have small integer roots. Once you have the polynomials, you might try the "rational root" theorem to narrow your choices.
 
  • #6
genericusrnme said:
unless you've got a nice matrix (read diagonal) you're going to have to use some tricks, if you're lucky you can use the tschirnhaus transformation but most likely you'll have to resort to numerical approximations for the eigenvalues (Newtons method or something) then you'll have to churn through to find the nullspace manually

it's pretty tedious work and chances are that you'd end up making an error doing it anyway..
you could just use mathematica or matlab, that'd be easier

if you don't have any of those, post your matrix and I'll give you the results, if you want?

Instead of MATLAB or mathematica, he/she might prefer to use Maple.

RGV
 
  • #7

1. What are eigenvalues and eigenvectors for a 5x5 matrix?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of a linear transformation on a vector space. In the case of a 5x5 matrix, the eigenvalues are the scalar values that, when multiplied by the eigenvectors, produce the same vector after transformation.

2. How do you calculate eigenvalues and eigenvectors for a 5x5 matrix?

To calculate the eigenvalues and eigenvectors for a 5x5 matrix, you can use a variety of methods such as the characteristic polynomial method, the power iteration method, or the QR algorithm. These methods involve finding the roots of the characteristic polynomial or repeatedly applying transformations until the desired eigenvectors are found.

3. Why are eigenvalues and eigenvectors important in 5x5 matrices?

Eigenvalues and eigenvectors are important in 5x5 matrices because they provide valuable information about the behavior of the matrix under linear transformations. They can also be used to diagonalize a matrix, making it easier to perform calculations and solve equations.

4. Can a 5x5 matrix have complex eigenvalues and eigenvectors?

Yes, a 5x5 matrix can have complex eigenvalues and eigenvectors. This is because complex numbers can be used as eigenvalues to describe the behavior of a matrix under certain transformations. In fact, a 5x5 matrix can have up to 5 complex eigenvalues and corresponding eigenvectors.

5. Are there any shortcuts or tricks for finding eigenvalues and eigenvectors of a 5x5 matrix?

There are some shortcuts and tricks for finding eigenvalues and eigenvectors of a 5x5 matrix, such as using the Cayley-Hamilton theorem or exploiting the symmetry of the matrix. However, these methods may not always work and it is important to have a solid understanding of the underlying concepts and equations involved in order to accurately find the eigenvalues and eigenvectors.

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