Matrices and eigenvalues. A comment in my answer.

In summary, the question is whether there are other matrices with eigenvalues 3 and 7, and if the set containing all these matrices is finite. The solution provided is that there are infinite matrices with these eigenvalues and different eigenvectors, and this can be proven by setting the elements of the matrix as variables in a linear system. Another way to prove this is by noting that a matrix with the elements 3 and 7 in the diagonal and any number in the off-diagonal has the same eigenvalues.
  • #1
sphlanx
11
0

Homework Statement



Hello and thanks again to anyone who has replied my posts. Your help is a great deal and really appreciated.

I have the following homework question which I have answered and I want a comment if it is valid or illogical:

We are given a matrix, with eigenvalues 3 and 7 respectively.

We are asked to say if there other matrices with the same eigenvalues and if the set containing all these matrices is finite.



Homework Equations


the matrix given:

2 1
-5 8

The Attempt at a Solution



I have thought of the following answer: My point is that there are infinite matrix with the same eigenvalues that have different eigenvectors. So I say that the linear system:

A(x1,y1)=λ1(x1,y1)
A(x2,y2)=λ2(x2,y2)

where A is a 2x2, has infinite solutions IF we take a11,a12,a21,a22(the elements of the matrix) as the variables of the linear system. The solutions will have x1,y1 and x2,y2 as constant parameters.

Have i got something terribly wrong?
 
Last edited:
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  • #2
No, that's perfectly correct.

Another way to prove that is to note that the matrix
[tex]\begin{bmatrix}3 & a \\ 0 & 7\end{bmatrix}[/tex]
has eigenvalues 3 and 7 for any a.
 

FAQ: Matrices and eigenvalues. A comment in my answer.

1. What is a matrix and how is it used in science?

A matrix is a rectangular array of numbers or variables arranged in rows and columns. It is commonly used in science to represent and manipulate data, equations, and relationships between variables.

2. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are properties of a square matrix that are used to understand and analyze its behavior. Eigenvalues represent the scaling factor of the eigenvectors when the matrix is multiplied by them.

3. How are matrices and eigenvalues used in data analysis and machine learning?

Matrices and eigenvalues are essential tools in data analysis and machine learning. They are used to represent and manipulate datasets, perform operations on data, and reduce the dimensionality of data for analysis.

4. What is the significance of the determinant of a matrix?

The determinant of a matrix is a scalar value that represents the scaling factor of the transformation represented by the matrix. It is used to determine if a matrix is invertible, solve systems of equations, and calculate the area/volume of a transformation.

5. How do matrices and eigenvalues relate to linear transformations?

Matrices and eigenvalues are used to represent and analyze linear transformations. The eigenvectors of a matrix represent the directions in which the transformation has no effect and the eigenvalues represent the scaling factor of the transformation in those directions.

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