Simple Matrix Help: Eigenvectors and Transformations

[phospho]

Question: http://gyazo.com/9c75baf06947bfa9f33a1772e6e6fc03I need help from b onwards

my answers to part a were)

lambda = 2, eigenvector of $$\begin{pmatrix} 1\\-1\\\end{pmatrix}$$

lambda = 4, eigen vector of $$\begin{pmatrix} 1\\1\\\end{pmatrix}$$

then for part b) I rearranged getting P^(-1)AP = D

I got orthogonal matrix P to be = $$k\begin{pmatrix} 1 & 1\\1 & -1\\\end{pmatrix}$$
where k = $\frac{1}{\sqrt{2}}$

then for D I got $$\begin{pmatrix} 4 & 0\\0 & 2\\\end{pmatrix}$$

so for part c) I done, a rotation of 45 degrees anticlockwise (about (0,0)), followed by a stretch of 4 parallel to x-axis, and x2 parallel to y-axis, followed by another anticlockwise rotation of 45 degrees.

However, in the answer they put 1. Rotation of pi/4 clockwise, 2. stretch, x4 || to x-axis, x2 || to y-axis, 3. rotation of pi/4 anticlockwise (about(0,0)).

as you can see this is different from me, and the only that I've done differently is got an eigenvector of $$\begin{pmatrix} 1\\-1\\\end{pmatrix}$$ instead of $$\begin{pmatrix} -1\\1\\\end{pmatrix}$$

as they did

any ideas where I went wrong

 

[Ray Vickson]

Question: http://gyazo.com/9c75baf06947bfa9f33a1772e6e6fc03


I need help from b onwards

my answers to part a were)

lambda = 2, eigenvector of $$\begin{pmatrix} 1\\-1\\\end{pmatrix}$$

lambda = 4, eigen vector of $$\begin{pmatrix} 1\\1\\\end{pmatrix}$$

then for part b) I rearranged getting P^(-1)AP = D

I got orthogonal matrix P to be = $$k\begin{pmatrix} 1 & 1\\1 & -1\\\end{pmatrix}$$
where k = $\frac{1}{\sqrt{2}}$

then for D I got $$\begin{pmatrix} 4 & 0\\0 & 2\\\end{pmatrix}$$

so for part c) I done, a rotation of 45 degrees anticlockwise (about (0,0)), followed by a stretch of 4 parallel to x-axis, and x2 parallel to y-axis, followed by another anticlockwise rotation of 45 degrees.

However, in the answer they put 1. Rotation of pi/4 clockwise, 2. stretch, x4 || to x-axis, x2 || to y-axis, 3. rotation of pi/4 anticlockwise (about(0,0)).

as you can see this is different from me, and the only that I've done differently is got an eigenvector of $$\begin{pmatrix} 1\\-1\\\end{pmatrix}$$ instead of $$\begin{pmatrix} -1\\1\\\end{pmatrix}$$

as they did

any ideas where I went wrong

Why would you think you are wrong? Your are NOT wrong! Any scalar multiple of an eigenvector is an eigenvector (for the same eigenvalue), so any column vector of the form [c,-c]^T (c ≠ 0) is an eigenvector for λ = 2. You take c = +1, the book takes c = -1. You could take c = -17/sqrt(423) if you wanted to; you would still have an eigenvector.

 

[phospho]

Why would you think you are wrong? Your are NOT wrong! Any scalar multiple of an eigenvector is an eigenvector (for the same eigenvalue), so any column vector of the form [c,-c]^T (c ≠ 0) is an eigenvector for λ = 2. You take c = +1, the book takes c = -1. You could take c = -17/sqrt(423) if you wanted to; you would still have an eigenvector.

but my rotations are different! This is why I assume I'm wrong somewhere, but I can't see where :(. In their rotations, they end back up to the original point (as well as the scale enlargement), I don't end up at the original point. I go 90 degrees away from it.

 

[vela]

Your matrix
$$ P = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$$ isn't just a rotation. It's a rotation and a reflection. If it were simply a rotation by 45 degrees, it should send (1,0) to (1/√2, 1/√2) and (0,1) to (-1/√2, 1/√2), but your P actually sends (0,1) to (1/√2,-1/√2).

 

[phospho]

Your matrix
$$ P = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$$ isn't just a rotation. It's a rotation and a reflection. If it were simply a rotation by 45 degrees, it should send (1,0) to (1/√2, 1/√2) and (0,1) to (-1/√2, 1/√2), but your P actually sends (0,1) to (1/√2,-1/√2).

oh okay, so would the order be

rotation pi/4 anticlockwise
reflect in y=x
enlargement of scale factor 4 and 2
rotation pi/4 anticlockwise
reflect in y=x?

 

[vela]

Yup.