# Eigenval/vect question, double check my answers, dont make sense

• imsleepy
In summary, to find the eigenvalues and eigenvectors of a matrix A, you can use the equation A-λI=0 and solve for λ to get the eigenvalues. To find the eigenvectors, substitute the eigenvalues into the equation (A-λI)v=0 and solve for v. When multiplying a matrix by a vector to check your answers, remember to multiply the rows of the matrix by the vector and add the terms.
imsleepy
eigenval/vect question, double check my answers, don't make sense....

## Homework Statement

find eigenvalues and eigenvectors of matrix A =
102
012
223

A -λI etc blah

## The Attempt at a Solution

I'm getting λ = 1, -1, 5.

For λ = 1, v = (-1,1,0)t (t being a real integer constant), and double checking says it's right.

For λ = -1, I'm getting v = (1,1,1)t but if i double check by multiplying it with the matrix A I'm not getting something in the form of v (i'm getting like.. 3,3,7, which doesn't make sense to me).

For λ = 5, i got v = (1,1,2)t but again double checking doesn't give a vector in this form! (multiplying this v by A gives me 3,3,14) :angry:

what am i doing wrong??

(1,1,1) is wrong, it is (-1,-1,1).
You do something wrong with matrix multiplication. How do you do it? The third eigenvector is correct.

ehild

ok yeah my bad, the evect for lambda = -1 is (-1,-1,1) (i rushed).

but multiplying A with (-1,-1,1) gives (-3,-3,7), which still doesn't make sense >:/

i get (1,1,-1) as you would expect

How do you do the multiplication?

Multiply (-1,-1,1) with the
first column of the original matrix: -1+0+2=1;
with the second column: 0-1+2=1;
with the third column: -2-2+3=-1.
The result is (1,1,-1) =-1(-1,-1,1) as it has to be.

ehild

ok i was multiplying column of matrix by each row of the vector, when i should be multiplying the rows of the matrix by the vector and adding terms.

It is the same, you can multiply each element of a row of the matrix with the corresponding element of the vector, and ADD them - so is it OK now?

ehild

A times (-1,-1,1) i (incorrectly) did:

1x-1 + 0x-1 + 2x-1
0x-1 + 1x-1 + 2x-1
2x 1 + 2x 1 + 3x 1

giving

-1-2
-1-2
2+2+3

resulting in (-3,-3,7)

which is incorrect.

i know how to do it now, thanks ehild and lanedance.
shouldve remembered this from years ago...

## 1. What is an eigenvalue and eigenvector?

An eigenvalue is a scalar value that represents how a linear transformation changes a vector. An eigenvector is a non-zero vector that is only scaled by the transformation, but its direction remains unchanged.

## 2. How are eigenvalues and eigenvectors used in science?

Eigenvalues and eigenvectors are used in many areas of science, including physics, engineering, and computer science. They are particularly useful in solving problems related to linear transformations, such as finding the principal axes of a system.

## 3. Can you explain the process of finding eigenvalues and eigenvectors?

The process of finding eigenvalues and eigenvectors involves solving a system of linear equations, where the coefficients of the equations are defined by the matrix representing the linear transformation. The eigenvalues are the solutions to the characteristic equation, and the eigenvectors are the corresponding solutions to the system of equations.

## 4. How do I check if my answers for eigenvalues and eigenvectors are correct?

To check if your answers for eigenvalues and eigenvectors are correct, you can use the eigenvalue equation to test your eigenvalues and corresponding eigenvectors. The equation should hold true for each eigenvalue and eigenvector pair.

## 5. What should I do if my answers for eigenvalues and eigenvectors don't make sense?

If your answers for eigenvalues and eigenvectors don't make sense, it's important to carefully review your calculations and make sure you haven't made any errors. You can also try using a different method to solve the problem or seek assistance from a colleague or instructor.

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