Find λ for Non-Uniform Scale & Shear Transformation

[Appleton]

Homework Statement

Find the two numerical values of λ such that

$\left(\begin{array}{cc}4&3\\1&2\end{array}\right) \left(\begin{array}{cc}u\\1\end{array}\right) =λ \left(\begin{array}{cc}u\\1\end{array}\right)$

Hence or otherwise find the equations of the two lines through the origin which are invariant under the transformation of the plane defined by

$\left(\begin{array}{cc}x\prime\\y\prime\end{array}\right) = \left(\begin{array}{cc}4&3\\1&2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right)$

Homework Equations

The Attempt at a Solution

I believe $\left(\begin{array}{cc}4&3\\1&2\end{array}\right)$ represents a non uniform scale and shear. If λ is a numerical value it can only scale uniformly which suggests there is no solution to the initial equation. However, my textbook tells me the 2 numerical values are 5 and 1.

I skipped this first part of the question and found the two invariant lines by setting $y=mx+c$ equal to $y\prime=mx\prime+c$. If someone could help me understand the first part of the question I would be very appreciative.

 

[BvU]

Hi App,

Can you write out the two equations that follow from the first matrix multiplication ? Eliminating $\lambda$ first seems the easiest to me; then the two values for u give the book values for $\lambda$.

 

[Ray Vickson]

Homework Statement

Find the two numerical values of λ such that

$\left(\begin{array}{cc}4&3\\1&2\end{array}\right) \left(\begin{array}{cc}u\\1\end{array}\right) =λ \left(\begin{array}{cc}u\\1\end{array}\right)$

Hence or otherwise find the equations of the two lines through the origin which are invariant under the transformation of the plane defined by

$\left(\begin{array}{cc}x\prime\\y\prime\end{array}\right) = \left(\begin{array}{cc}4&3\\1&2\end{array}\right) \left(\begin{array}{cc}x\\y\end{array}\right)$

Homework Equations

The Attempt at a Solution

I believe $\left(\begin{array}{cc}4&3\\1&2\end{array}\right)$ represents a non uniform scale and shear. If λ is a numerical value it can only scale uniformly which suggests there is no solution to the initial equation. However, my textbook tells me the 2 numerical values are 5 and 1.

I skipped this first part of the question and found the two invariant lines by setting $y=mx+c$ equal to $y\prime=mx\prime+c$. If someone could help me understand the first part of the question I would be very appreciative.

The question is asking you to find the eigenvalues $\lambda_1, \lambda_2$ of the matrix
$$A =\pmatrix{4&3\\1&2}$$
and to show that the eigenvectors of $A$ have the form
$$\pmatrix{u\\1}$$
See, eg., http://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx .

 

[HallsofIvy]

The equation $\begin{pmatrix}4 & 3 \\ 1 & 2 \end{pmatrix}\begin{pmatrix} u \\ 1 \end{pmatrix}= \lambda\begin{pmatrix} u \\ 1 \end{pmatrix}$ is the same as the two equations $4u+ 3= \lambda u$ and $u+ 2= \lambda$. That second equation is the same as $u= \lambda- 2$. Replace u in the first equation with $\lambda- 2$ to get an equation in $\lambda$.

 

[Appleton]

Thanks for the help. I guess I wasn't taking into account the fact that u is not constant.

I am still having difficulty understanding how the 2nd part of the question follows from the first. My book has not yet explicitly introduced me to eigenvectors, so I am presuming that the connection can be deduced from very rudimentary matrix principles.

 

[Ray Vickson]

Thanks for the help. I guess I wasn't taking into account the fact that u is not constant.

I am still having difficulty understanding how the 2nd part of the question follows from the first. My book has not yet explicitly introduced me to eigenvectors, so I am presuming that the connection can be deduced from very rudimentary matrix principles.

In fact, u is a constant---it is just the case that you don't know its value yet. Ditto for $\lambda$.

Basically, you just need to write down the two equations in $u$ that you get from rows 1 and 2 of your matrix; then you have two equations in the single unknown $u$. (The equations have an undetermined parameter $\lambda$ in them.) In order for these two equations to be consistent, the value (or values) of $\lambda$ must be special, and figuring out these special values is the crux of your problem.

 

[Mark44]

Thanks for the help. I guess I wasn't taking into account the fact that u is not constant.

I am still having difficulty understanding how the 2nd part of the question follows from the first. My book has not yet explicitly introduced me to eigenvectors, so I am presuming that the connection can be deduced from very rudimentary matrix principles.

Yes, it can, which should reinforce what an eigenvector is when it is formally defined.

 

[Appleton]

I think I get it now. So the value of λ is superfluous for the second part of the question, it is only really u that we are concerned with at this stage, since u defines the vectors which, when multiplied by λ define the 2 invariant lines through the origin.

Thanks for all your help.