Eigenvector Calculations and Verification for y1+y2=5

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In summary, The conversation discusses the confirmation of an answer and the use of eigenvalues and trace in calculating a quantity for triangular matrices. The characteristic equation is used to find the eigenvalues, and the sum of the eigenvalues is equal to the trace of the matrix. W|A is then used to confirm the results.
  • #1
ertagon2
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Can you please confirm my answer?
View attachment 7539
det(yI-A) = 0
(y-1)(y-4)=0
y1+y2=5
 

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  • #2
Yes, it is correct.

Incidentally, did you already learn about the eigenvalues of triangular matrices?
Did you perhaps already learn about the trace of a matrix (= the sum along the main diagonal)?

This could help you to calculate the required quantity without having to set up the characteristic polynomial. (Of course, that is not much work here, either.)
 
  • #3
I get the characteristic equation:

\(\displaystyle x^2-(1+4)x+(1\cdot4-0\cdot3)=x^2-5x+4=(x-1)(x-4)=0\)

Hence:

\(\displaystyle \lambda_1+\lambda_2=1+4=5\quad\checkmark\)

W|A confirms our results:

W|A - eigenvalues for ((1,0),(3,4))
 

1. What is an eigenvector?

An eigenvector is a vector in a given vector space that remains unchanged when multiplied by a particular matrix. It is a special type of vector that represents the direction of the transformation caused by the matrix.

2. What does the term "eigen" mean in "eigenvector"?

The term "eigen" comes from the German word "eigen" which means "own" or "inherent." In mathematics, an eigenvector is a vector that is inherent to a particular matrix and remains unchanged when multiplied by that matrix.

3. What is the significance of eigenvectors in linear algebra?

Eigenvectors are important in linear algebra because they provide a way to simplify complicated matrix operations and understand the behavior of linear transformations. They also have applications in fields such as physics, engineering, and computer science.

4. How do you find eigenvectors?

To find eigenvectors, you first need to find the eigenvalues of the matrix. Then, for each eigenvalue, you solve a system of equations to find the corresponding eigenvector. This can be done using methods such as Gaussian elimination or by using software like MATLAB or Python.

5. What is the difference between an eigenvector and a characteristic vector?

There is no difference between an eigenvector and a characteristic vector. They both refer to the same type of vector that remains unchanged when multiplied by a particular matrix. However, the term "eigenvector" is more commonly used in mathematics, while "characteristic vector" is more commonly used in physics and engineering.

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