So I was originally given the equation Q(x)= (x1)^2 + 4(x1)(x2) + 4(x2)^2. I put

Question

So I was originally given the equation Q(x)= (x1)^2 + 4(x1)(x2) + 4(x2)^2.

I put this into a matrix with column 1={1,2} and column 2= {2,4}.

I then diagonalized this, finding P to be a 2x2 matrix with column 1= u1={1/sqrt(5), 2/sqrt()5} and column 2= u2 = {-2/sqrt(5), 1/sqrt()5}

Eigenvalues are 5 and 0, so matrix D in A=PDP-1 has column 1 = {5,0} and column 2={0,0}

I then did a change of variable, assuming x=Py, by saying that:

xT Ax = (Py)T A(Py).

I came to find that xT Ax = yT Dy = 5(y1)^2

Where x = {x1,x2} and y={y1,y2}.

This is where I've hit a wall. I'm asked to sketch xT Ax = 5 (y1)^2 and clearly label the principal axes, but I don't know how to go about doing this and the book doesnt explain well. Can anyone help me out here? There's an example showing that some vector x was chosen and plugged into y=(P-inverse)x and then y was plugged into the equation corresponding toxT Ax = 5 (y1)^2, but I don't know how the values for the vector x were chosen. Any insight on how to do this? Because I am simply at a loss.

Thank you for any and all of your help!

Explanation / Answer

P is of the form
cos(t) sin(t)
-sin(t) cos(t)
which is a rotation clockwise of angle cos(t)=1/5, so t=arcos(1/5)

So y moves to x with such a rotation

Then x moves to y to a counterclockwise rotation of angle t=arcos(1/5)

P-1 is be of the form

cos(t) -sin(t)

sin(t) cos(t)

"but I don't know how the values for the vector x were chosen"

the orthonormal basis (1,0), (0,1) will give you the new basis y=P-1x .Joining the origin with these points will give you the new axes.

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