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In summary, the video explains how to find the solution for the equation $$\left( \begin{array}{cc} 8 & 1 \\ -8 & -1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = 0$$with the given solutions (1, -1) and (1, -8). It is mentioned that there are infinite possible solutions and that the order of the values in the solution is arbitrary. The presenter demonstrates how to find the solution and explains the concept of eigenvectors.
The equation he obtains is
$$\left( \begin{array}{cc} 8 & 1 \\ -8 & -1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = 0$$
##(x,y) = (1, -8)## is a solution, but not ##(x,y) = (-8, 1)##. Try it for yourself: substitue the possible solutions in the above equation, and see what works.

By the way, there is an infinite number of possible solutions: any vector ##a (1, -8)## is also a solution (with ##a## a scalar). Which one to choose is arbitrary. In the video, he could have chosen ##y=1## instead, and found ##(x,y) = (-1/8, 1)## (you can check for yourself that solution also).

1 person
You talk here, and in a similar question in the homework section as if the "eigenvector" were just two numbers in arbitrary order. It is not- it is a vector which, in this case, can be represented[/b] as an ordered pair of numbers.

You should, yourself, have done the multiplication
$$\begin{pmatrix}8 & 1 \\ -8 & -1\end{pmatrix}\begin{pmatrix}-8 \\ 1\end{pmatrix}= \begin{pmatrix}-64+ 1 \\ 64- 1\end{pmatrix}= \begin{pmatrix}-63 \\ 63\end{pmatrix}$$
which is NOT
$$0\begin{pmatrix}-8 \\ 1 \end{pmatrix}$$

1 person

1. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a square matrix, produces a multiple of itself. It represents a direction in which there is no change when the matrix is applied.

2. What is the significance of eigenvectors?

Eigenvectors are important in many areas of mathematics and science, particularly in linear algebra and quantum mechanics. They are used to solve systems of linear equations, determine stability of a system, and understand the behavior of quantum systems.

3. What is the order of eigenvectors?

The order of eigenvectors refers to the sequence in which they are listed or calculated. It is important to specify the order when working with multiple eigenvectors, as it can affect the outcome of calculations.

4. How do you calculate the order of eigenvectors?

The order of eigenvectors is determined by the number of columns in the matrix they are associated with. For example, a 3x3 matrix will have 3 eigenvectors, while a 4x4 matrix will have 4 eigenvectors.

5. How does the order of eigenvectors affect the results of calculations?

The order of eigenvectors can affect the results of calculations, particularly when working with matrices with repeated eigenvalues. In these cases, the order of eigenvectors can determine which linear combination of eigenvectors is used to represent a certain eigenvalue, and thus impact the final solution.

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