Abstract
Let \(\mathbb{k}\) be an arbitrary field, V a finite-dimensional vector space over \(\mathbb{k}\), and F: V → V a linear endomorphism of V over \(\mathbb{k}\). We call a pair (F, V ) a space with operator or just an operator over \(\mathbb{k}\). Given two spaces with operators (F 1, U 1) and (F 2, U 2), a linear map C: U 1 → U 2 is called a homomorphism of spaces with operators if F 2 ∘ C = C ∘ F 1, or equivalently, if the diagram of linear maps
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Notes
- 1.
See Problem 7.7 on p. 168.
- 2.
See Sect. 7.3 on p. 164.
- 3.
See Proposition 3.8 on p. 53.
- 4.
See Proposition 5.1 on p. 103.
- 5.
See Sect. 14.3 on p. 351.
- 6.
See Definition 14.4 on p. 345.
- 7.
See Sect. 3.2 on p. 46.
- 8.
See Sect. 7.3 on p. 164.
- 9.
That is, the monic polynomial μ F (t) of lowest positive degree such that μ F (F) = 0 (see Sect. 8.1.3 on p. 175).
- 10.
See formula (8.13) on p. 182.
- 11.
Which says that χ F (F) = 0.
- 12.
Or cyclic basis.
- 13.
Recall that we write ν i for the length of the ith row in the Young diagram ν and write ν t for the transposed Young diagram. Thus, ν m t means the length of the mth column of ν.
- 14.
Or completely reducible.
- 15.
See Sect. 10.5 on p. 244.
- 16.
See Sect. 10.1 on p. 237.
- 17.
Counted without multiplicities.
- 18.
Note that this agrees with Corollary 15.5 on p. 367.
- 19.
That is, without multiple roots.
- 20.
Or an idempotent operator.
- 21.
See Sect. 15.2.1 on p. 367.
- 22.
That is, the maximal integer m such that \((t-\lambda )^{m} \in \mathop{\mathcal{E}\ell}\nolimits (F)\) (see Sect. 15.1.5 on p. 365).
- 23.
That is, power series converging everywhere in \(\mathbb{C}\).
- 24.
Compare with Example 4.4 on p. 81.
- 25.
See formula (15.9) on p. 375.
- 26.
It is highly edifying for students to realize this program independently using the standard topologies, in which the convergence of functions means the absolute convergence over every disk in \(\mathbb{C}\), and the convergence of matrices means the convergence with respect to the distance on \(\mathrm{Mat}_{n}(\mathbb{C})\) defined by | A, B | = ∥ A − B ∥ , where \(\Vert C\Vert \stackrel{\text{def}}{=}\max _{v\in \mathbb{C}^{n}\setminus 0}\vert Cv\vert /\vert v\vert \) and \(\vert w\vert ^{2}\stackrel{\text{def}}{=}\sum \vert z_{i}\vert ^{2}\) for \(w = (z_{1},z_{2},\ldots,z_{n}) \in \mathbb{C}^{n}\). A working test for such a realization could be a straightforward computation of \(e^{J_{n}(\lambda )}\) directly from the definitions.
- 27.
See Sect. 15.2.3 on p. 370.
- 28.
See Definition 15.2 on p. 379.
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Gorodentsev, A.L. (2016). Linear Operators. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_15
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