# Easy math

can someone look at the image...
can someone teach me/ help me remember how we can get
e11 and e12
thank you so much
tuan

here is the picture

#### Attachments

• untitled.PNG
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cristo
Staff Emeritus
What is this; a homework question or what? I don't really understand what you want explaining to you. Is there a specific question that the attachment is a solution to?

CompuChip
Homework Helper
You can get e11 and e12 as in the picture. Just write down the eigenvalue equation, multiply it out and solve the two given equations for the two unknowns. That's all that is happening. If you want more help, you need to be more specific.

HallsofIvy
Homework Helper
Apparently this is to find eigenvectors of the matrix
$$\left(\begin{array}{cc} 3 & \sqrt{2} \\ \sqrt{2} & 4\end{array}\right)$$
corresponding to eigenvalue 5.

Since you want, by definition of "eigenvalue" and "eigenvector"
$$\left(\begin{array}{cc} 3 & \sqrt{2} \\ \sqrt{2} & 4\end{array}\right)\left(\begin{array}{c}e_{11} \\ e_{12}\end{array}\right)= \left(\begin{array}{c} 5e_{11} \\ 5e_{12}\end{array}\right)$$

$$\left(\begin{array}{cc}-2e_{11}+ \sqrt{2}e_{12}\\ \sqrt{2}e_{11}- e_{12}\end{array}\right)= \left(\begin{array}{c}0 \\ 0\end{array}\right)$$

That gives you the two equations $-2e_{11}+ \sqrt{2}e_{12}= 0$ and $\sqrt{2}e_{11}- e_{12}= 0$. Because 5 is an eigenvalue, if you try to solve those two simultaneous equations you will find they are "dependent"- they reduce to a single equation: $\sqrt{2}e_{11}= e_{12}$. Choosing either of those to be any number you want, you can solve for the other. In your attachment, they choose $e_{11}= 1$ so $e_{12}= \sqrt{2}$. Any eigenvector of the equation, corresponding to eigenvalue 5, is a multiple of $(1, \sqrt{2})$.

The other eigenvalue, by the way, is 2. What are the eigenvectors corresponding to eigenvalue 2?

CompuChip
In your attachment, they choose $e_{11}= 1$ so $e_{12}= \sqrt{2}$. Any eigenvector of the equation, corresponding to eigenvalue 5, is a multiple of $(1, \sqrt{2})$.
Actually, they choose $e_{11}, e_{12}$ such that the eigenvector has unit length
$$|| (e_{11}, e_{12}) || = \sqrt{ e_{11}^2 + e_{12}^2 }$$