Abstract
In the previous chapters, we have defined some numbers associated to a matrix, such as the determinant, trace, and rank. In this chapter, we focus on scalars and vectors known as eigenvalues and eigenvectors. The eigenvalues and eigenvectors have many important applications, in particular, in the study of differential equations.
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Notes
- 1.
We can deduce here that two different eigenvalues of f cannot have the same associated eigenvector u.
- 2.
See Definition 1.2.5.
- 3.
This formula holds only for commuting matrices.
- 4.
As this exercise shows, the minimal polynomial provides another criterion for diagonalizability.
- 5.
The goal of this exercise is to exhibit the beautiful unity of the solutions of the quadratic and cubic equations, in a form that is easy to remember, which is based on the circulant matrices. This exercise is based on a result in [12]
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Said-Houari, B. (2017). Eigenvalues and Eigenvectors. In: Linear Algebra. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-63793-8_7
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