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Let $\boldsymbol{x}=\left(\begin{matrix}x\\ y\end{matrix}\right)$ be a vector in two-dimensional real space. By finding the eigenvalues and eigenvectors of $\boldsymbol{M}$, sketch the locus of points $\boldsymbol{x}$ that satisfy $$ \boldsymbol{x^TMx}=4$$ given that $$\boldsymbol{M}=\left(\begin{matrix}&5 &\sqrt{3}\\ &\sqrt{3} &3\end{matrix}\right). $$

I found two eigenvalues to be $\lambda_1 = 6$ and $\lambda_2=2$, and the corresponding eigenvectors are $$ \boldsymbol{v}_1=\left(\begin{matrix}\sqrt{3}\\ 1\end{matrix}\right)\quad\text{ and }\quad \boldsymbol{v}_2=\left(\begin{matrix}1\\ -\sqrt{3}\end{matrix}\right)$$ (if I'm not mistaken :) ), but... what now? Frankly, I can't figure out how to make this helpful to find $\boldsymbol{x^TMx}=4$.

Any hints?

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  • $\begingroup$ The eigenvalues are $5\pm\sqrt3$, not $6,2$ because $6+2\ne5+5$ $\endgroup$ – Shubham Johri Dec 17 '18 at 10:44
  • $\begingroup$ @ShubhamJohri see the edited question. I wrongly typed one entry of $M$. $\endgroup$ – VanDerWarden Dec 17 '18 at 10:45
  • $\begingroup$ Yes, now the eigenvalues and eigenvectors are fine $\endgroup$ – Shubham Johri Dec 17 '18 at 10:47
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The eigenvectors are orthogonal and span $\Bbb R^2$. This means $\mathbf v=\begin{bmatrix}x\\y\end{bmatrix}=c_1\mathbf x_1+c_2\mathbf x_2$.

$\mathbf v^TM\mathbf v=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)M(c_1\mathbf x_1+c_2\mathbf x_2)\\=(c_1\mathbf x_1^T+c_2\mathbf x_2^T)(c_1\lambda_1\mathbf x_1+c_2\lambda_2\mathbf x_2)\\=c_1^2\lambda_1||\mathbf x_1||^2+c_2^2\lambda_2||\mathbf x_2||^2\\=24c_1^2+8c_2^2=4$

$\implies6c_1^2+2c_2^2=1$

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Some hints (as you requested):

  • What we're studying is a second degree equation in $x$ and $y$, so it's a conic section curve (ellipse, hyperbola, or parabola).

  • After we find the eigenvalues and eigenvectors, we can use them to diagonalize the matrix $\mathbf{M}$.

  • The diagonalization process is really just a change of coordinate system, say from $(x,y)$ coordinates to $(u,v)$ coordinates

  • In $(u,v)$ coordinates, since we're now dealing with a diagonal matrix, the given equation takes the form $au^2 + bv^2 = 1$. This is a conic section curve whose geometry you probably understand.

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  • $\begingroup$ Not op but I got $3u^2+2v^2=8$. So do I just plot this elipse with axis being the two eigenvectors? And how long shoud $1$ unit be? Say for axis1 with $v_1$, should $1$ unit be normal 1 unit, normal 6 units (eigenvalue) or 1 unit should be $2$ normal units (length of $v1$). I think the last one is correct $\endgroup$ – Anvit Dec 17 '18 at 11:11
  • $\begingroup$ @Anvit: Yes, the eigenvectors define the directions of the axes of symmetry of the conic. If $\mathbf v$ is an eigenvector, then so is $k\mathbf v$, for any $k \ne 0$, so the eigenvectors can't tell you anything about scale. $\endgroup$ – bubba Dec 17 '18 at 11:17
  • $\begingroup$ @bubba -- there's little or no mileage in appealing to downvoters to explain or justify their objections. It's best not to betray that you heed them atall. $\endgroup$ – AmbretteOrrisey Dec 17 '18 at 11:35

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