Find the eigenvalues and the corresponding eigenvectors of A. $$A=\begin{bmatrix} 4 & -3 \\ 2 & -1\end{bmatrix}$$

Not looking for an answer but rather some direction. I do not know what an eigenvalue entirely is and cannot seem to comprehend it from my textbook, so anything can help me at this point. Anything is greatly appreciated. Thank you, Matt.

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    $\begingroup$ It's unlikely that any short answer here will really help you. I'm sorry your book confuses you. Perhaps search for "what is an eigenvalue" and find a link that explains things in terms that make sense. You could begin with wikipedia. Your other questions here (with answers you haven't accepted) suggest that you try to read another textbook from the beginning. $\endgroup$ – Ethan Bolker Dec 11 '17 at 17:05

A good place to start understanding these concepts on a more intuitive level is this video series, but I'll give a brief outline here of what the concepts mean.

Eigenvectors of a matrix are vectors which, when transformed by the matrix, are scaled by a constant. Eigenvalues are the constants by which they are scaled.

So if I have a matrix that rotates a vector $30^\circ$ around the x-axis, its only eigenvector is $\langle1, 0, 0\rangle$, and the corresponding eigenvalue is $1$.

You can find tons of explanations of how to actually calculate the eigenthings of a matrix just by some googling, so I'll leave that to you.

Does this explanation help?

Also, for the matrix you gave specifically, it has eigenvectors $\langle3, 2\rangle$ and $\langle1, 1\rangle$ with eigenvalues $2$ and $1$ respectively. This means that any scalar multiple of $\langle3, 2\rangle$ will be doubled when transformed by the matrix, and any scalar multiple of $\langle1, 1\rangle$ will be unchanged.

  • $\begingroup$ yes, thank you. $\endgroup$ – Matt Spahr Dec 11 '17 at 17:17

To find the eigenvalues $λ_1,λ_2$ of your $2\times 2$ matrix, you need to solve the following equation :

$$\det(A-λI)=0 \Rightarrow \bigg| \begin{matrix} 4-λ & -3 \\ 2 & -1-λ\end{matrix} \bigg|=0 \Rightarrow \dots$$

the solutions $λ_1,λ_2$ that you will get from $λ$ from solving the equation above, are the eigenvalues of your given matrix.

Now, to find the corresponding eigenvectors $v_1,v_2$ of each eigenvalues, you simply have to solve the following system of equations :

$$(A-λ_1I)v_1 = 0$$

$$(A-λ_2I)v_2 = 0$$

the vectors $v_1,v_2$ that you will eventually calculate, will be the eigenvectors of your given matrix.

To read up on more about eigenvalues/eigenvectors, check out this link here.

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    $\begingroup$ I don't think that will help if you don't explain how that equation comes about. For people who just started on the topic, it might be more useful to start from $Av=\lambda v$ to help them get the idea better. $\endgroup$ – Karn Watcharasupat Dec 11 '17 at 17:26
  • $\begingroup$ Using this, I have found eigenvalues of 1 and 2, does this sound correct? $\endgroup$ – Matt Spahr Dec 11 '17 at 17:48
  • $\begingroup$ @MattSpahr Yes ! These are the correct eigenvalues. $\endgroup$ – Rebellos Dec 11 '17 at 17:53

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