Abstract
This final chapter on Frobenius’ mathematics is devoted to the paper he submitted to the Berlin Academy on 23 May 1912 with the title “On matrices with nonnegative elements” [231].
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Notes
- 1.
In what follows, such matrices A will be called nonnegative, and this property will be indicated with the notation A ≥ 0. Similarly, A > 0 will mean that all the elements of A are positive.
- 2.
The present chapter is based upon my paper [278], which contains more details, especially regarding the work of Perron and Markov.
- 3.
For details on Perron’s life and family background see [166].
- 4.
The information about Pringsheim is based on Perron’s memorial essay [470].
- 5.
- 6.
- 7.
See Satz II [467, p. 12], which establishes more than I have stated below in Theorem 17.5.
- 8.
For an exposition of this historic proof, see [278, Appendix 6.1].
- 9.
Perron gave only a proof of (1), from which (2) and (3) follow easily. Regarding his proof of (1), see [278, p. 675, n. 15].
- 10.
Let M, a, b, … be the distinct absolute values of the roots \(\rho _{0},\rho _{1},\ldots,\rho _{n}\). Then the numerator of (17.27) can be expressed as \({M}^{\nu +1}\alpha _{\nu +1} + {a}^{\nu +1}\beta _{\nu +1} + {b}^{\nu +1}\gamma _{\nu +1} + \cdots \) , where each of \(\alpha _{\nu +1},\beta _{\nu +1},\gamma _{\nu +1}\) is a sum of at most n + 1 complex numbers of absolute value 1. Thus each of \(\alpha _{\nu +1},\beta _{\nu +1},\gamma _{\nu +1}\) has absolute value at most n + 1. Likewise, the denominator of (17.27) is expressible as \({M}^{\nu }\alpha _{\nu } + {a}^{\nu }\beta _{\nu } + {b}^{\nu }\gamma _{\nu } + \cdots \), with each of \(\alpha _{\nu },\beta _{\nu },\gamma _{\nu }\) having absolute value at most n + 1. Thus the ratio in (17.27) that is under the limit operation, on division of the numerator and denominator by M ν, is
$$\displaystyle{\frac{M\alpha _{\nu +1} + a{(a/M)}^{\nu }\beta _{\nu +1} + b{(b/M)}^{\nu }\gamma _{\nu +1} + \cdots } {\alpha _{\nu } + {(a/M)}^{\nu }\beta _{\nu } + {(b/M)}^{\nu }\gamma _{\nu } + \cdots }.}$$Clearly, all the terms in the numerator except the first approach 0 as ν → ∞. Likewise, all the terms in the denominator except the first approach 0 as ν → ∞. Thus, assuming the limit ρ′ exists in (17.27), we see that \(\rho ^{\prime} =\lim _{\nu \rightarrow \infty }M(\alpha _{\nu +1}/\alpha _{\nu }) =\lim _{\nu \rightarrow \infty }({M}^{\nu +1}\alpha _{\nu +1})/({M}^{\nu }\alpha _{\nu })\), which is to say that only the roots of absolute value M contribute to the limiting value ρ′.
- 11.
The “known theorem” was presumably that increasing (respectively, decreasing) sequences of real numbers that are bounded above (respectively, below) converge.
- 12.
It does not seem that Perron was seeking a purely algebraic proof of his theorem in the sense of a proof that was completely free of propositions from analysis. For example, he never expressed a similar dissatisfaction with his proof of his seminal Lemma 17.6, despite the fact that it repeatedly invoked basic theorems from analysis such as the intermediate value theorem for continuous functions [278, Appendix 6.1]. The intermediate value theorem was also invoked in Frobenius’ proof of Perron’s theorem. It should also be noted that although Frobenius took up the challenge of a determinant-based proof of Perron’s theorem, as a student of Weierstrass, he was not averse to employing results from complex analysis, notably Laurent expansions, in his proofs of theorems about matrices (as in Sections 7.5.1, 7.5.5, and16.1.5).
- 13.
In proving Lemma 17.6, Perron could have used the weaker induction hypothesis Adj (ρ 0 I − A) > 0 but did not, thereby unnecessarily complicating his proof; see [278, Appendix 6.1].
- 14.
- 15.
Thus, e.g., in the limit, Adj (ρ 0 I − A) > 0 for A > 0 becomes Adj (ρ 0 I − A) ≥ 0 for A ≥ 0. In particular, the (α, α) entry of Adj (ρ 0 I − A), namely \(\varphi _{\alpha \alpha }(\rho _{0})\), is nonnegative; and since by (17.28), \(\varphi ^{\prime}(\rho _{0})\) is the sum of all the \(\varphi _{\alpha \alpha }(\rho _{0})\), it follows that \(\varphi ^{\prime}(\rho _{0}) \geq 0\) with \(\varphi ^{\prime}(\rho _{0}) = 0\) only if all \(\varphi _{\alpha \alpha }(\rho _{0})\) vanish, thereby giving the above necessary condition for ρ 0 to be a multiple root.
- 16.
Two such present-day applications—Markov chains and input–output-type economic analysis—existed at the time of Frobenius’ work on positive and nonnegative matrices (1908–1912). Markov introduced the eponymous chains in 1908 (see Section 17.3), and the mathematician Maurice Potron announced an economic theory analogous to input–output analysis in 1911 [487, 488] with details given in 1913 [489]. Although Frobenius was apparently unaware of these developments, it is of interest to note that neither Markov nor Potron ascribed an importance to nonnegative characteristic vectors in their respective applications. For further general information about Potron, whose work remained unappreciated until recently, see [1, 17, 18]. Note also that Wilfried Parys is working on an annotated Potron bibliography and on historical aspects of Perron–Frobenius theory in economics.
- 17.
A principal minor determinant of ρ 0 I − A of degree n − k is one obtained by deleting the same k rows and columns of ρ 0 I − A, e.g., by deleting the first k rows and the first k columns.
- 18.
What Frobenius used, without any explanation, was the fact that if B is any matrix and Adj (B) = (β i j ), then detB = 0 implies \(\beta _{ij}\beta _{ji} =\beta _{ii}\beta _{jj}\) for all i ≠ j. Since when \(B =\rho _{0}I - A\), β ii and β j j are positive (being principal minors), it follows from \(\beta _{ij}\beta _{ji} =\beta _{ii}\beta _{jj}\) that β i j and β ji are not just nonnegative but positive. The identity \(\beta _{ij}\beta _{ji} =\beta _{ii}\beta _{jj}\) follows from a very special case of a well-known identity due to Jacobi [13, p. 50]. It also follows readily from the more basic identity \(B \cdot \mathrm{Adj}\,(B) = (\det B)I\), which when detB = 0 implies rankAdj (B) ≤ 1, and so all 2 ×2 minors of Adj (B) must vanish. (Viewed in modern terms, B ⋅Adj (B) = 0 means that the range of Adj (B) is contained in the null space of B. When 0 is a simple root of B, this means that rankAdj B ≤ 1.)
- 19.
Presumably for this reason, Theorem 17.16 is not stated by Frobenius as a formal theorem, although it is alluded to in his prefatory remarks. The proof is given on pp. 549–550 of [231].
- 20.
In terms of the graph-theoretic characterization of irreducibility given above in a footnote to Frobenius’ official definition, G(A) is a directed n-cycle and so connected.
- 21.
- 22.
See e.g., pp. 569, 571 and 576 of the English translation by Petelin. In what is to follow, all page references will he to this translation (cited in the bibliographic reference for Markov’s 1908 paper [431]).
- 23.
If x is a characteristic vector for ρ and m = max i | x i | , let i 0 be such that \(\vert x_{i_{0}}\vert = m\). Then the i 0th equation of ρI = P x is \(\rho x_{i_{0}} =\sum _{ j=1}^{n}p_{i_{0}j}x_{j}\). Taking absolute values and using the triangle inequality implies \(\vert \rho \vert m \leq \sum _{j=1}^{n}p_{i_{0}j}\vert x_{j}\vert \leq \big (\sum _{j=1}^{n}p_{i_{0}j}\big)m = 1 \cdot m\), whence | ρ | ≤ 1. Markov sought variations on this line of reasoning that would prove | ρ | < 1 for all roots ρ ≠ 1 for P satisfying certain conditions [278, §6.3].
- 24.
See [278, p. 694], where these conditions are denoted by (ID) and (IID) and Markov’s actual words are quoted.
- 25.
- 26.
Potron spoke of “partially reduced” matrices [487, p. 1130], by which he meant the equivalent of reducible matrices.
- 27.
The directed graph G(P) contains the 4-cycle 1 → 3 → 2 → 4 → 1 and so is connected.
- 28.
On the founding of the institute, see [22, pp. 148–153].
- 29.
Complete reducibility for A ≥ 0 means that A is permutationally similar to a block diagonal matrix in which the diagonal blocks are irreducible. Cf. Theorem 17.24 above.
- 30.
Write \(A = {S}^{-1}JS\), where J is the Jordan canonical form of A. The above-described properties of the characteristic roots of A imply that \(\lim _{n\rightarrow \infty }{J}^{n} = J_{\infty } = \mathrm{Diag.\,Matrix}(1, 0,\ldots, 0)\). Thus \(\lim _{n\rightarrow \infty }{A}^{n}v_{0} = {S}^{-1}J_{\infty }Sv_{0}\) exists. Of course, in von Mises’ theorem, A is assumed to be symmetric, so that J is diagonal and J ∞ = Diag. Matrix(1, 0, …, 0) is easier to see.
- 31.
- 32.
See, e.g., the paper by Hadamard and Fréchet [258, p. 2083], where von Mises’ work is called to the reader’s attention and praised. Hadamard and Fréchet also state (on p. 2083) that von Mises (among others mentioned) did his work without knowledge of Markov’s paper [431]. Although the basis for this statement is uncertain, it seems to be based on their belief that Markov’s work was available only in Russian, whereas, as noted earlier, a German translation had been available since 1912 in the German edition of Markov’s book [432].
- 33.
For further information on Romanovsky, see [164].
- 34.
Judging by his remark [502, p. 267], Romanovsky was the first to use the term “stochastic matrix.” For him it meant (i) P ≥ 0 (ii) with row sums equaling 1 and (iii) no zero column. Nowadays, condition (iii) is not usually included in the definition of a stochastic matrix, and I have not included this condition in my references to stochastic matrices.
- 35.
- 36.
The most interesting and historically significant part of Romanovsky’s paper is the concluding paragraphs, where he responded to Kaucký’s criticism by attempting to characterize those P which admit all primitive kth roots of unity, k ≥ 3, as characteristic roots. These paragraphs are of interest because they involved what turns out to be an alternative characterization of the degree of imprimitivity k of an irreducible matrix, a characterization that has a graph-theoretic interpretation (A is cyclic of index k). Romanovsky himself made no reference to the theory of graphs, and it is doubtful he was thinking in such terms, since his ideas were motivated by the well-known determinant-theoretic formula for the coefficients of the characteristic polynomial \(\varphi (r) = \vert rI - A\vert \), as is evident from his subsequent, more detailed papers [503, p. 215] and [504, p. 163].
- 37.
In 1937, Gantmacher and Krein [241] had already used Perron’s Lemma 17.6 as proved by Frobenius in 1908 to develop their theory of strictly positive (respectively nonnegative) matrices—n ×n matrices such that all k ×k minors are positive (respectively, nonnegative) for all k = 1, …, n. Such matrices arise in the mechanical analysis of small oscillations. See [242] for a comprehensive account.
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Hawkins, T. (2013). Nonnegative Matrices. In: The Mathematics of Frobenius in Context. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6333-7_17
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