Finding eigenvectors of general matrix given char. eqn.

In summary: As Ray showed, the characteristic equation only gives you the eigenvalues. You cannot determine the eigenvectors without knowing the matrix A. In summary, You cannot determine the eigenvectors of a matrix just by knowing the characteristic equation. The characteristic equation only gives the eigenvalues of the matrix, not the eigenvectors. To find the eigenvectors, you need to know the matrix A.
  • #1
gtmcph
2
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Homework Statement

This is my first post, so forgive me if anything's out of order.
Assumean operator A satisfies the following equation:
1+2A-A^2-2A^3=0
Find the eigenvalues and eigenvectors for A



Homework Equations





The Attempt at a Solution


So the eigenvalues are +1,-1, and -1/2. At least those are the values which satisfy the characteristic equation above. What I don't know is how to find the associated eigenvectors (without being given a concrete matrix). Is there a better way to solve this than making up a general 3x3 matrix and muddling through the (A-λI)v=λv equations? Thanks.
 
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  • #2
gtmcph said:

Homework Statement

This is my first post, so forgive me if anything's out of order.
Assumean operator A satisfies the following equation:
1+2A-A^2-2A^3=0
Find the eigenvalues and eigenvectors for A



Homework Equations





The Attempt at a Solution


So the eigenvalues are +1,-1, and -1/2. At least those are the values which satisfy the characteristic equation above. What I don't know is how to find the associated eigenvectors (without being given a concrete matrix). Is there a better way to solve this than making up a general 3x3 matrix and muddling through the (A-λI)v=λv equations? Thanks.

You can't find the eigenvectors just given the characteristic equation. They depend on the exact matrix A.
 
  • #3
gtmcph said:

Homework Statement

This is my first post, so forgive me if anything's out of order.
Assumean operator A satisfies the following equation:
1+2A-A^2-2A^3=0
Find the eigenvalues and eigenvectors for A



Homework Equations





The Attempt at a Solution


So the eigenvalues are +1,-1, and -1/2. At least those are the values which satisfy the characteristic equation above. What I don't know is how to find the associated eigenvectors (without being given a concrete matrix). Is there a better way to solve this than making up a general 3x3 matrix and muddling through the (A-λI)v=λv equations? Thanks.

In general, you cannot say for sure that ##p(x) = (1/2) + x - (1/2) x^2 - x^3## is the characteristic polynomial of the matrix A. You are told that p(A) = 0, so p is an annihilating polynomial of A, but one can imagine there are matrices in which the characteristic polynomial C(x) is a divisor of p(x). For example, if A is a matrix with C(x) = 1-x^2 (that is, with ##A^2 = I##, then we also have p(A) = 0, since ##p(x) = (x + 1/2)C(x)##, so
[tex] p(A) = \left( A + \frac{1}{2}I \right) C(A) = 0,[/tex] but p is not the characteristic polynomial. In that example, A is a 2×2 matrix, and only +1 and -1 are eigenvalues!

However, if A is 3×3 then p(x) is the characteristic polynomial, and the eigenvalues are, indeed, ±1 and -1/2.
 
  • #4
Thanks Ray, that's a good point. But the main question is how can we find general eigenvectors given nothing but this equation which the operator satisfies? This question was posed by my physics professor, so he seems convinced that it is indeed possible. Any ideas?
 
  • #5
gtmcph said:
Thanks Ray, that's a good point. But the main question is how can we find general eigenvectors given nothing but this equation which the operator satisfies? This question was posed by my physics professor, so he seems convinced that it is indeed possible. Any ideas?
Ray's point is the same as Dick's: No, you cannot determine the eigenvectors from the characteristic equation.
 

FAQ: Finding eigenvectors of general matrix given char. eqn.

What is an eigenvector?

An eigenvector is a vector that, when multiplied by a specific matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains the same after the matrix multiplication, but the magnitude is scaled by a constant factor.

What is a characteristic equation?

A characteristic equation is a polynomial equation used to find the eigenvalues of a matrix. The coefficients of this equation are determined by the entries of the matrix, and the roots of the equation are the eigenvalues.

How do you find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you first need to set up and solve the characteristic equation by taking the determinant of the matrix and setting it equal to zero. Once you have the eigenvalues, you can use them to find the corresponding eigenvectors.

What is the difference between a left eigenvector and a right eigenvector?

A left eigenvector is a row vector that, when multiplied by a matrix, results in a scalar multiple of itself on the left side. A right eigenvector is a column vector that, when multiplied by a matrix, results in a scalar multiple of itself on the right side. In other words, a left eigenvector is multiplied on the left side of the matrix, while a right eigenvector is multiplied on the right side.

Can a matrix have multiple eigenvectors for the same eigenvalue?

Yes, a matrix can have multiple eigenvectors for the same eigenvalue. This is because eigenvectors are only defined up to a scalar multiple, so any multiple of an eigenvector is also an eigenvector for the same eigenvalue.

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