Eigenvalues and -vectors in class

In summary, the speaker is a student in a math class and was assigned to give a presentation on eigenvalues and eigenvectors. They explain that they have to define the concept and provide an explanation, as well as demonstrate the use of eigenvalues with exercises and an application. The speaker also mentions considering using a geometrical approach to present the topic. When questioned about their knowledge of eigenvalues, the speaker explains that they are only 16 but have a strong understanding of math and are assigned to give the presentation as part of their course.
  • #1
nonequilibrium
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Hi,

For math we were assigned a subject which we'd present during one class' hour in a group. My group got "Eigenvalues & eigenvectors". So basically first I have to give the definition and explain what it actually is (AX = [tex]\lambda[/tex]X) and then we can spend the rest of the 45 min on making class exercises on this new subject and (something I think welcome to eigenvalues) showing an application of it, to make it less abstract.

Any ideas of how I could present this?

I read somewhere it's used for the Schrödinger equation -- a very interesting piece of science, but I don't think that's something you "show to the aid of eigenvalues".

I was thinking, maybe I could start by drawing a two-dimensional plane with an x and y-axis and lead to eigenvalues from the geometrical point of view
 
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  • #2
Do you even know what "eigenvalues" are? If not, why would you agree to give a class talk on them?
 
  • #3
Oh, I'm only 16, but I do 8h of math, meaning it's my main course and as an assignment, the class was divided into groups of three. And as we're currently seeing matrices, each group was assigned a different aspect of it (like eigenvalues for us), which we would learn about for ourselves (we could ask our teacher for help) and then we each get an hour to teach it to the rest of the class.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important concepts in linear algebra. Eigenvalues are scalars that represent the scale factor by which an eigenvector is stretched or compressed when a linear transformation is applied to it. Eigenvectors are non-zero vectors that do not change their direction when a linear transformation is applied to them, but may be scaled by the corresponding eigenvalue.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by solving the characteristic equation of a matrix. The characteristic equation is obtained by setting the determinant of the matrix minus a scalar lambda to zero. The resulting values for lambda are the eigenvalues, and the corresponding eigenvectors can be found by solving the system of linear equations obtained by substituting the eigenvalues into the matrix equation.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important in many fields, including physics, engineering, and computer science. They are used to represent fundamental properties of linear transformations and can be used to simplify complex systems of equations. In addition, they are used in data analysis and machine learning to identify patterns and reduce dimensionality.

4. Can a matrix have multiple eigenvalues and eigenvectors?

Yes, a matrix can have multiple eigenvalues and corresponding eigenvectors. The number of eigenvalues is equal to the dimension of the matrix, and each eigenvalue may have multiple eigenvectors associated with it. In some cases, a matrix may have repeated eigenvalues, resulting in fewer distinct eigenvectors.

5. How are eigenvalues and eigenvectors used in real-life applications?

Eigenvalues and eigenvectors are used in a variety of real-life applications, such as image compression, pattern recognition, and data analysis. In image compression, they are used to represent the underlying structure of an image and eliminate redundant information. In pattern recognition, they are used to identify similarities between patterns and classify data. In data analysis, they can be used to reduce the dimensionality of a dataset and extract important features for further analysis.

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