Evaluating a wrong value of Eigenvector

[zak100]

Homework Statement

I think I am evaluating a wrong value for eigen vector. Wrong in the sense that first row is giving different value and next row is giving a different value.
The matrix is:
[0 1]
[-2 -3]
My eigen values are λ=-1 & λ=-2. I get the fine vectors for λ=-1 but λ=-2 x vanishes.

Homework Equations

A-λI = 0
& Ax = λx

The Attempt at a Solution

[0 1] * [x] = -2 * [x]
[-2 -3] [y] [y]

First row gives: y = -2x -2y
2x = 3y
Now x = 2/3 y

μ =[2, 3] (instead of column, i am showing row vector)

But the second row gives me: -2x -3y = -2x -2y (therfore x vanishes)

they have evaluated
μ=[1, -2] (instead of column, i am showing row vector)

Some body please guide me.

Zulfi.

 

[fresh_42]

Homework Statement

I think I am evaluating a wrong value for eigen vector. Wrong in the sense that first row is giving different value and next row is giving a different value.
The matrix is:
[0 1]
[-2 -3]
My eigen values are λ=-1 & λ=-2. I get the fine vectors for λ=-1 but λ=-2 x vanishes.

Homework Equations

A-λI = 0
& Ax = λx

The Attempt at a Solution

[0 1] * [x] = -2 * [x]
[-2 -3] [y] [y]

First row gives: y = -2x -2y
2x = 3y
Now x = 2/3 y

μ =[2, 3] (instead of column, i am showing row vector)

But the second row gives me: -2x -3y = -2x -2y (therfore x vanishes)

they have evaluated
μ=[1, -2] (instead of column, i am showing row vector)

Some body please guide me.

Zulfi.

Both eigenvalues are correct, but I don't follow your solution. It seems you first deal with $\lambda = -2$. For this we have
$\left( \begin{bmatrix}0&1\\-2&-3\end{bmatrix} - (-2) \cdot \begin{bmatrix}1&0\\0&1\end{bmatrix} \right) \cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$
which results in $2x+y=0$ and $-2x-y=0$ which is different from what you wrote.

 

[zak100]

Hi,
Thanks for your response.
For evaluating, eigen vectors, I used the equation
Ax = λx

Is it right or not?

Zulfi.

 

[fresh_42]

Hi,
Thanks for your response.
For evaluating, eigen vectors, I used the equation
Ax = λx

Is it right or not?

Zulfi.

Yes and yes. Yes, the equation itself $A\vec{x}=\lambda \vec{x}$ is right, and yes, that's not what you used.

This equation is actually two equations.
1. row: $0\cdot x + 1 \cdot y = (-2) \cdot x$
2. row: $-2 \cdot x +(-3)\cdot y = (-2) \cdot y$

 

[zak100]

Hi,
I have shown my calculations. But if you are not following and i can do it on a paper and post it as a file. Is it fine?

May be you are right because I am getting the same prob in another question also.

Zulfi.

 

[fresh_42]

[0 1] * [x] = -2 * [x]
[-2 -3] [y] [y]

This does not make sense.

First row gives: y = -2x -2y
2x = 3y
Now x = 2/3 y

This is wrong.

 

[zak100]

Hi,

RHS of row1 is not correct. I am getting : -2x -2y
because λ * x is:
-2 * [ x]
[y]
So its : -2x -2y.
In your case its just -2 *x. I can't understand this because if you agree with this:
A* x = λ *x

RHS = λ * x
& λ = -2
& X = [x]
[y]

Please guide me. Thanks for your time.

Zulfi.

 

[fresh_42]

Hi,

RHS of row1 is not correct. I am getting : -2x -2y
because λ * x is:
-2 * [ x]
[y]
So its : -2x -2y.
In your case its just -2 *x. I can't understand this because if you agree with this:
A* x = λ *x

RHS = λ * x
& λ = -2
& X = [x]
[y]

Please guide me. Thanks for your time.

Zulfi.

You have to understand how matrix multiplication goes. The equation you have is actually matrix (A) times vector (x,y in column) = eigenvalue (-2) [times identity matrix (I), which is not written] times vector (x,y in column). Now matrix multiplication is always row times column for short. This means it is
$$
\begin{bmatrix}a&b\\c&d\end{bmatrix} \cdot \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}a\cdot x + b \cdot y \\ c\cdot x + d \cdot y \end{bmatrix}
$$
This means for our case that we have
$$
\begin{bmatrix}0&1\\-2&-3\end{bmatrix} \cdot \begin{bmatrix}x\\y\end{bmatrix}= \begin{bmatrix}0\cdot x + 1\cdot y\\ (-2)\cdot x+ (-3)\cdot y\end{bmatrix} = (-2)\cdot \begin{bmatrix}1&0\\0&1\end{bmatrix} \cdot \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}-2 & 0 \\0 &-2 \end{bmatrix} \cdot \begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}(-2)\cdot x + 0\cdot y \\ 0\cdot x + (-2)\cdot y\end{bmatrix}
$$
and now you can compare
$$
\begin{bmatrix}0\cdot x + 1\cdot y\\ (-2)\cdot x+ (-3)\cdot y\end{bmatrix} =\begin{bmatrix}y\\-2x-3y\end{bmatrix} = \begin{bmatrix}(-2)\cdot x + 0\cdot y \\ 0\cdot x + (-2)\cdot y\end{bmatrix}= \begin{bmatrix}-2x\\-2y\end{bmatrix}
$$
row by row.

 

[Mark44]

Hi,

RHS of row1 is not correct. I am getting : -2x -2y
because λ * x is:
-2 * [ x]
[y]
So its : -2x -2y.
In your case its just -2 *x. I can't understand this because if you agree with this:
A* x = λ *x

RHS = λ * x
& λ = -2
& X = [x]
[y]

Please guide me. Thanks for your time.

Zulfi.

You're making this much harder than it really is.
Instead of working directly with $Ax = \lambda x$, use $(A - \lambda I)x = 0$
Here, $A = \begin{bmatrix}0 & 1 \\ -2 & -3 \end{bmatrix}$, and $\lambda = -2$, so $A - \lambda I = \begin{bmatrix}2 & 1 \\ -2 & -1 \end{bmatrix}$.
The last matrix is equivalent to $\begin{bmatrix}2 & 1 \\ 0 & 0 \end{bmatrix}$ (which can be obtained by very simple row reductionn), so the matrix equation reduces to 2x + y = 0. We can arbitrarily set y = 1, so x = -1/2. That is, $\begin{bmatrix} -1/2 \\ 1 \end{bmatrix}$.

All eigenvectors for $\lambda = -2$ for matrix A are some multiple of this vector.

 

[zak100]

Hi,
Thanks I would try by using Identity matrix & redo all my examples.

This means I would use:
Ax =λ*I * x.

This would correct my multiplication.

Zulfi.

 

[fresh_42]

Hi,
Thanks I would try by using Identity matrix & redo all my examples.

This means I would use:
Ax =λ*I * x.

This would correct my multiplication.

Zulfi.

You can as well simply write $(-2)\cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-2x\\-2y\end{bmatrix}$ which is the same.