Eigenvalues and Eigenvectors of 3x3 matricies

In summary: Perhaps it helps if you post an example of a matrix and your result for the characteristic equation where you have doubts?Cubic equations, in general, are difficult. If the equation has integer coefficients, you can try to use the 'rational root theorem': if m/n is a a rational root of the polynomial equation, ax^n+ \cdot\cdot\cdot+ bx+ c= 0, with integer coefficients, then the denominator, n, must divide the leading coefficient, a, and the numerator, m, must divide the constant term, c. You can factor those two cofficients to find all possible rational roots, then put them into the equation
  • #1
9_lulu_0
1
0
Hello

Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues.

Can someone please help..
Thanks
 
Physics news on Phys.org
  • #2
There is usually nothing fancy about it, you just need some way to solve the cubic equation. Often it is possible to guess one solution, divide out a factor and solve the resulting quadratic equation. Otherwise there may be other tricks available, or you might need some general formula (the degree 3 equivalent of the quadratic "abc" formula).

Perhaps it helps if you post an example of a matrix and your result for the characteristic equation where you have doubts?
 
  • #3
Cubic equations, in general, are difficult.

If the equation has integer coefficients, you can try to use the 'rational root theorem': if m/n is a a rational root of the polynomial equation, [itex]ax^n+ \cdot\cdot\cdot+ bx+ c= 0[/itex], with integer coefficients, then the denominator, n, must divide the leading coefficient, a, and the numerator, m, must divide the constant term, c. You can factor those two cofficients to find all possible rational roots, then put them into the equation to see if they work.

Of course, it is possible that a cubic equation does not have any rational roots. In that case, you would have to use "Cardano's cubic formula"

I don't see how we can say more without seeing the specific matrix or cubic equation you are talking about.
 
Last edited by a moderator:
  • #4
Try diagonalizing the matrix... Look for similarity transformation... But it is not easier than characteristic polynomial anyway. But at least you stay with matrices instead of polynomialshence, a little bit more fun.
 
  • #5
Try factoring the determinant into linear factors?
 
  • #6
There's a couple of neat properties when finding eigenvalues that might help. For one, the trace of your matrix is equal to the sum of your eigenvalues. The determinant is equal to the product of the eigenvalues as well.
 
  • #7
9_lulu_0 said:
Hello

Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues.

Can someone please help..
Thanks
So give us some examples of polynomial equations you are having troulbe with.
 

Related to Eigenvalues and Eigenvectors of 3x3 matricies

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of a square matrix. Eigenvalues are scalar values that represent how a matrix stretches or compresses a particular eigenvector in a given direction.

Eigenvectors are the corresponding vectors that do not change direction when multiplied by the matrix, only their magnitude changes by a factor of the eigenvalue.

2. What are the applications of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors have various applications in different fields such as physics, chemistry, and engineering. In physics, they are used to describe the behavior of quantum mechanical systems. In chemistry, they are used to analyze molecular vibrations. In engineering, they are used to solve problems related to structural mechanics, signal processing, and control systems.

3. How do you find the eigenvalues and eigenvectors of a 3x3 matrix?

To find the eigenvalues and eigenvectors of a 3x3 matrix, we first need to calculate the characteristic polynomial of the matrix. Then, we solve the polynomial to find the eigenvalues. Next, we substitute each eigenvalue into the original matrix and solve the system of equations to find the corresponding eigenvectors.

4. Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors. This happens when the matrix has complex entries or when the characteristic polynomial has complex roots. In this case, the eigenvectors will also have complex entries.

5. How do eigenvalues and eigenvectors relate to matrix diagonalization?

Eigenvalues and eigenvectors play a crucial role in matrix diagonalization. A square matrix can be diagonalized if and only if it has a full set of linearly independent eigenvectors. In other words, the matrix can be represented as a diagonal matrix with the eigenvalues on the main diagonal.

Similar threads

I Find the Eigenvalues and eigenvectors of 3x3 matrix
    • Linear and Abstract Algebra
Replies
12
Views
1K
I Eigenvalues and Eigenvectors question
    • Linear and Abstract Algebra
Replies
16
Views
2K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
    • Linear and Abstract Algebra
Replies
3
Views
2K
Can quaternion group be represented by 3x3 matricies?
    • Linear and Abstract Algebra
Replies
11
Views
2K
I Ladder operators and SU(2) representation
    • Linear and Abstract Algebra
Replies
1
Views
2K
I Eigenvalue as a generalization of frequency
    • Linear and Abstract Algebra
Replies
4
Views
1K
Eigenvalues/eigenvectors using householder and QR
    • Linear and Abstract Algebra
Replies
5
Views
4K
How to find the eigenvector for a perturbated Hamiltonian?
    • Advanced Physics Homework Help
Replies
13
Views
1K
I General Method for Mapping an Ellipsoid to Unit Sphere
    • Linear and Abstract Algebra
Replies
1
Views
959
I Eigenvectors for degenerate eigenvalues
    • Linear and Abstract Algebra
Replies
7
Views
8K

Hot Threads

Recent Insights

Back
Top