Simple Yes or No Will Do Eigenvectors

In summary, an eigenvector is a vector that remains unchanged in direction but may change in magnitude when multiplied by a specific square matrix. To calculate eigenvectors, the equation Av = λv must be solved, where A is the matrix, v is the eigenvector, and λ is the scalar multiple. Eigenvectors are significant in fields such as physics, engineering, and computer science for studying linear transformations. A matrix can have multiple eigenvectors associated with different eigenvalues, and these are used in the process of diagonalization to simplify calculations and solve linear equations.
  • #1
essie52
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Simple Yes or No Will Do... Eigenvectors

Homework Statement



I think my prof made a mistake when writing this problem:

Find a basis of the space V of all 3 x 3 matrices A for which the vectors <1, 1> and <1, 2> are eigenvectors and thus determine the dimensions of V.

Is this problem possible? Wouldn't the matrices have to be 2 x 2 given the vectors?
 
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  • #2


It sounds like a mistake; the eigenvectors of a 3x3 matrix would each be of dimension 3.
 

1. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a specific square matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains unchanged after the matrix transformation, but its magnitude may change.

2. How do you calculate eigenvectors?

The process of finding eigenvectors involves solving the equation Av = λv, where A is the square matrix, v is the eigenvector, and λ is the scalar multiple. This can be done by finding the eigenvalues first, then using them to find the corresponding eigenvectors.

3. What is the significance of eigenvectors?

Eigenvectors are important in many fields, including physics, engineering, and computer science. They are used to study and understand systems that involve linear transformations, such as systems of differential equations, quantum mechanics, and data analysis.

4. Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors associated with different eigenvalues. However, there can only be as many distinct eigenvalues as the dimension of the matrix, and the eigenvectors must be linearly independent.

5. How do eigenvectors relate to the concept of diagonalization?

Diagonalization is the process of finding a diagonal matrix that is similar to the original matrix. Eigenvectors are used in this process to transform the original matrix into a diagonal matrix. This is useful for simplifying calculations and solving systems of linear equations.

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