Thread: Find a vector given a matrix

1. Find a vector given a matrix

We did a problem in class I didn't understand.

"Find a vector w so that:"
[1 3]w = 2w
[1 -1]

I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...

2. Re: Find a vector given a matrix

Originally Posted by MrJank
We did a problem in class I didn't understand.
"Find a vector w so that:"
[1 3]w = 2w
[1 -1]
Use $w = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]$

3. Re: Find a vector given a matrix

Originally Posted by Plato
Use $w = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]$
The equality is a false statement. Any vector will be an nx1 matrix. We know n=2. The multiplication will yield a 1x1 matrix, which will never be equal to a scalar times a 2x1 matrix.

4. Re: Find a vector given a matrix

Originally Posted by MrJank
We did a problem in class I didn't understand.

"Find a vector w so that:"
[1 3]w = 2w
[1 -1]

I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...
Let w be the vector
$\displaystyle w = \left [ \begin{matrix} a \\ b \end{matrix} \right ]$

Then we need to find a, b such that the following is true:
$\displaystyle \left [ \begin{matrix} 1 & 3 \\ 1 & -1 \end{matrix} \right ] ~ \left [ \begin{matrix} a \\ b \end{matrix} \right ] = 2 \left [ \begin{matrix} a \\ b \end{matrix} \right ]$

Multiplying this out we get the two equations
$\displaystyle \begin{matrix} a + 3b = 2a \\ a - b = 2b \end{matrix}$

What can a and b be?

-Dan

5. Re: Find a vector given a matrix

Originally Posted by SlipEternal
The equality is a false statement. Any vector will be an nx1 matrix. We know n=2. The multiplication will yield a 1x1 matrix, which will never be equal to a scalar times a 2x1 matrix.
?????
$\left[ {\begin{array}{*{20}{c}} 1&3 \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {x + 3y} \\ {x - y} \end{array}} \right] \\\\\left[ {\begin{array}{*{20}{c}} 1&3 \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}} {0} \\ {0} \end{array}} \right]$

6. Re: Find a vector given a matrix

It appears that the solution is any vector [w1, w2] where w1/w2=3.

If I am not mistaken, this kind of solution often occurs in vibrations where multiple masses oscillate with any amplitude, however these amplitudes have fixed ratios (eigenvectors). Been a while though, I may be wrong.

7. Re: Find a vector given a matrix

Originally Posted by Plato
?????
$\left[ {\begin{array}{*{20}{c}} 1&3 \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {x + 3y} \\ {x - y} \end{array}} \right] \\\\\left[ {\begin{array}{*{20}{c}} 1&3 \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}} {0} \\ {0} \end{array}} \right]$
I thought the matrix was just the vector [1 3]. I didn't realize the [1 -1] was part of it.

8. Re: Find a vector given a matrix

Originally Posted by MrJank
We did a problem in class I didn't understand.

"Find a vector w so that:"
[1 3]w = 2w
[1 -1]

I'm confident I could do it if it equaled only 2, but the 2w is throwing me off...
If it only equaled two it would be impossible! The left side is a vector while "2" is not.
Write [tex]w= \begin{bmatrix} x \\ y \end{bmatrix}[tex]. The equation becomes
$\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x+ 3y \\ x- y \end{bmatrix}= 2\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2x \\ 2y \end{bmatrix}$.

That is the same as the two equations x+ 3y= 2x and x- y= 2y. It should be easy to see that the only solution is x= y= 0.

A little more "sophisticated" would be to write the equation as $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$.

Then $\displaystyle \begin{bmatrix}1 - 2 & 3- 0 \\ 1- 0 & -1- 2\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}-1 & 3 \\ 1 & -3 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.

Now, as long as that matrix is invertible, we could multiply both sides by its inverse and get, since the product of any matrix with the 0 vector is the 0 vector, $\displaystyle \begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.

9. Re: Find a vector given a matrix

Originally Posted by HallsofIvy
If it only equaled two it would be impossible! The left side is a vector while "2" is not.
Write [tex]w= \begin{bmatrix} x \\ y \end{bmatrix}[tex]. The equation becomes
$\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}x+ 3y \\ x- y \end{bmatrix}= 2\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2x \\ 2y \end{bmatrix}$.

That is the same as the two equations x+ 3y= 2x and x- y= 2y. It should be easy to see that the only solution is x= y= 0.

A little more "sophisticated" would be to write the equation as $\displaystyle \begin{bmatrix}1 & 3 \\ 1 & -1 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}2 & 0 \\ 0 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$.

Then $\displaystyle \begin{bmatrix}1 - 2 & 3- 0 \\ 1- 0 & -1- 2\end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}-1 & 3 \\ 1 & -3 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.

Now, as long as that matrix is invertible, we could multiply both sides by its inverse and get, since the product of any matrix with the 0 vector is the 0 vector, $\displaystyle \begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$.
Actually, $w=c\begin{bmatrix}3\\1\end{bmatrix}$ where $c$ is any real number. It is finding an eigenvector for the matrix with eigenvalue 2.

10. Re: Find a vector given a matrix

Originally Posted by MrJank
"Find a vector w so that:"
[1 3]w = 2w
[1 -1]...
Originally Posted by SlipEternal
Actually, $w=c\begin{bmatrix}3\\1\end{bmatrix}$ where $c$ is any real number. It is finding an eigenvector for the matrix with eigenvalue 2.
Actually this is a poorly formed question and even more poorly presented one.
It does say find a vector...

11. Re: Find a vector given a matrix

Originally Posted by Plato
Actually this is a poorly formed question and even more poorly presented one.
It does say find a vector...
I agree. My point was not that $x=y=0$ is wrong. I was pointing out that it is not the only solution.