Abstract
We have seen that a necessary and sufficient condition for a matrix to be diagonalizable is that a set of its eigenvectors forms a basis of the underlying linear space. In the first half of this chapter, we study the Jordan normal form and the Jordan decomposition which generalize the above fact for arbitrary matrices not necessarily diagonalizable. We explain how to compute them in Python for large matrices which may be hard and cumbersome using only paper and pencil. We also make a program which generates classroom or examination problems.
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Notes
- 1.
This equality means that both sides are equal as linear map**s because the right-hand side is not a matrix but a linear map**.
- 2.
In general, \(\boldsymbol{A}^*\boldsymbol{A}\) is a positive semi-definite matrix, but in almost all cases it is a positive definite matrix because \(\boldsymbol{A}\) is randomly generated. The spectrum of a positive definite matrix does not contain the origin.
- 3.
Remark that a positive matrix is completely different from a positive definite matrix with a similar name. The positivity is not invariant under similarity nor under unitary equivalence of matrices. In this sense it is a peculiar property of matrices. Positive matrices are very important for applications.
- 4.
In almost all cases this is a positive matrix.
- 5.
This argument is based on the so-called \(\varepsilon \)-\(\delta \) definition of a limit.
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© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
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Tsukada, M., Kobayashi, Y., Kaneko, H., Takahasi, SE., Shirayanagi, K., Noguchi, M. (2023). Jordan Normal Form and Spectrum. In: Linear Algebra with Python. Springer Undergraduate Texts in Mathematics and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-99-2951-1_8
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DOI: https://doi.org/10.1007/978-981-99-2951-1_8
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