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7. The Cayley–Hermite Problem and Matrix Algebra

verfasst von : Thomas Hawkins

Erschienen in: The Mathematics of Frobenius in Context

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Abstract

Less than 2 years after Frobenius submitted his “monograph” on the problem of Pfaff, he submitted another monograph [181], in which he showed that bilinear forms in n variables (or equivalently, their coefficient matrices) form a linear associative algebra that can be represented symbolically and used to great advantage in solving linear algebraic problems. He was motivated to do so by a problem, called here the Cayley–Hermite problem, which had hitherto been treated on the generic level.

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Vorheriges Kapitel The Problem of Pfaff

Nächstes Kapitel Arithmetic Investigations: Linear Algebra

Much of this chapter is based on my paper [270], where I first called attention to the historical importance of this problem.

In this respect, Cayley was astutely following the lead of Lagrange, whose treatment of the principal axes theorem 1775 (Section 4.4.1) involved such a replacement, which Cauchy then generalized to n variables in 1829 (Section 4.4.2).

For any invertible matrix M, ${M}^{-1} = 1/(\det M) \cdot {[\mathrm{Cof}(M)]}^{t}$, where Cof(M) denotes the matrix of cofactors, i.e., the (i, j) entry of Cof(M) is the cofactor corresponding to m _ij. See the discussion following (4.15).

The letter is located in the library of St. John’s College, Cambridge, along with others that passed between Cayley and Sylvester. I am grateful to Dr. I. Grattan-Guinness for calling my attention to the existence of these letters, to Prof. E. Koppelman for informing me of their content, and to the Masters and Fellows of St. John’s College for permission to make the above quotation.

Special cases of the theorem were presented by Hamilton in the context of his theory of quaternions [262, p. 567], but that theory does not appear to have been a major influence upon Cayley. Cayley himself denied any such influence in a polemical exchange with Tait [349, pp. 153, 164]. The absence of any mention of quaternions in his letter of 19 November 1857 supports his denial, as does our discussion of the role of the Cayley–Hermite problem.

If it did, then since all its characteristic roots are 0, we would have $L = SM{S}^{-1}$, where detS ≠ 0, since M is the Jordan canonical form for this situation. But then ${L}^{2} = S{M}^{2}{S}^{-1} = 0\not =M$.

The same is true of Sylvester’s subsequent treatment of this problem, as I have shown in [270, §6].

Nowadays, Laguerre is remembered primarily because of the polynomials that bear his name. For a broader perspective on his life and work, see [16, 482].

Rosanes was later M. Born’s teacher at Breslau. According to Born, it was by recalling Rosanes’ lectures of 1903 on algebra and analytic geometry that he recognized the connection between Heisenberg’s new approach to quantum mechanics and matrix algebra. See Jammer [318, p. 204], who, however, incorrectly states that Rosanes was Frobenius’ student, although, as the following discussion indicates, Rosanes undoubtedly read, and was taught by, Frobenius’ 1878 paper.

That P ^t A P = A follows by elementary matrix algebra, which Rosanes, regrettably, lacked.

This follows from the more general result given by Frobenius that for a family u A + v A ^t, any elementary divisors of the form (u + v)^2κ or ${(u - v)}^{2\kappa +1}$ occur doubled [181, p. 364, I]. (There is a typographical error in Frobenius’ paper (p. 22) that is only partially corrected in the collected works (p. 364).)

Frobenius used the notation P ^′ to denote the transpose of P.

Frobenius denoted the identity matrix by the letter E (for Einheit).

As Frobenius showed [181, §§1–2], p(A) and q(A) commute, and so p(A) and [q(A)]⁻¹ commute.

Quoted in Section 5.6.3.

For example, the canonical form $J = J_{3}(2) \oplus J_{3}(1/2) \oplus J_{2}(-1)$ has elementary divisors in accord with Theorem 7.9. Hence all Rosanes transformations with elementary divisors (r − 2)³, ${(r - 1/2)}^{3}$, and (r + 1)², are given by $P = LJ{L}^{-1}$, where L is any invertible 7 ×7 matrix. To determine a bilinear form left invariant by P, use the fact that in general, P ^t is similar to P, and by Theorem 7.9, P is similar to P ^− 1, which is similar to J ^− 1. Thus P ^t is similar to J ^− 1, and so ${P}^{t} = M{J}^{-1}M$ for some invertible M. It then follows that P leaves invariant the form $F(x,y) = {x}^{t}Ay$, $A = M{L}^{-1}$.

The above quotation is from Hermite’s paper in Crelle’s Journal [288, p. 309], which is restricted to ternary forms. German mathematicians, including Frobenius, were not familiar with Hermite’s solution for quadratic forms in n variables [287], published as it was in an obscure British journal. The n-variable solution contains nothing that was not already implied by the solution in the ternary case.

The formula for T says that $S(I - U) = T(I + U)$, which may be rewritten as $SU + TU = S - T$, i.e., $U = {(S + T)}^{-1}(S - T)$.

This follows by writing Q in block form as $Q = \left (\begin{array}{ccc} q_{1}q &\cdots &q_{n}q\\ \end{array} \right )$, from which it follows from ${q}^{t}(\mathrm{Adj}\,S)q = 1$ that Q(Adj S)Q = Q and so $U_{0}^{2} = I_{n}$.

Thus, e.g., $U = \left (\begin{array}{cccc} - 1 & 1 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 1 \\ 0 & 0 & 0 & - 1 \end{array} \right )$, which has (r + 1)² as a repeated elementary divisor and satisfies U ^t SU = S for $S = \left (\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 4 & - 1 & 2 \\ 0 & - 1 & 0 & 0 \\ 1 & 2 & 0 & 3 \end{array} \right )$, is a solution not covered by generalizing the formulas of Bachmann and Hermite.

Although Frobenius did not mention it, since det(I + U _h) → 0 implies that the coefficients of U will be bounded for $\vert h\vert \leq \varepsilon$, the Bolzano–Weierstrass theorem, which he surely knew, would imply that a sequence h _n → 0 exists such that $U_{0} =\lim _{n\rightarrow \infty }U_{h_{n}}$ exists.

As noted earlier, Frobenius did not use block multiplication of matrices. He knew that his symbols represented linear substitutions and coefficient matrices as well as bilinear forms, but the form interpretation was favored and so he worked with direct sums of forms to achieve the same results.

In detail: $U = {G}^{-1}JG =\lim _{h\rightarrow 0}\{{G}^{-1}{(\tilde{S}+\tilde{T}_{h})}^{-1}[{({G}^{t})}^{-1}{G}^{t}](\tilde{S}-\tilde{T}_{h})G\} = {[{G}^{t}(\tilde{S}+\tilde{T}_{h})G]}^{-1}\times {G}^{t}(\tilde{S}-\tilde{T}_{h})G = {(S+T_{h})}^{-1}(S-T_{h})$.

This is (7.4), except T has been replaced by − T because Frobenius dealt with it in that (equivalent) form.

For every $x$, $\Vert Rx\Vert = \sqrt{Rx \cdot Rx} = \sqrt{x \cdot {R}^{t } Rx} = \sqrt{x \cdot x} =\Vert x\Vert$. If a is a characteristic root of R and $x\not =0$ is such that $Rx = ax$, then the above implies $\vert a\vert \cdot \Vert x\Vert =\Vert Ux\Vert =\Vert x\Vert$, and so | a | = 1.

Recall from the discussion surrounding (4.36) that k is the largest order of a pole at r = a of a coefficient $\varphi _{ij}(r)/\varphi (r)$ of $\mathrm{Adj}(rI - R)/\det (rI - R) = {(rI - R)}^{-1}$.

Since ${(rI - R)}^{-1} =\mathrm{ Adj}(rI - R)/\det (rI - R)\; \mathop{=}\limits \mathrm{def}\;\varphi _{ij}(r)/\varphi (r)$, all the coefficients $\varphi _{ij}(r)/\varphi (r)$ have poles of order at most 1 at r = a, i.e., if (r − a)^q divides $\varphi (r)$, then ${(r - a)}^{q-1}$ divides $\varphi _{ij}(r)$ for all i, j. This means that D _n − 1(r), the polynomial greatest common divisor of all the $\varphi _{ij}(\lambda )$, is also divisible by ${(r - a)}^{q-1}$; and so $\varphi (r)/D_{n-1}(r) = E_{n}(r) =\psi (r)$, the minimal polynomial of R, is the product of distinct linear factors. But from Weierstrass’ definition of elementary divisors, $E_{n}(r) =\prod _{ i=1}^{d}{(r - a_{i})}^{e_{i}}$, where a ₁, …, a _d are the distinct characteristic roots of R and ${(r - a)}^{e_{i}}$ is the elementary divisor for a _i of maximal exponent. Hence all e _i equal 1, and all elementary divisors are linear.

Frobenius used his calculational skill to derive (7.28) from (7.25). First be multiplied both sides of (7.25) by $rR = {[{r}^{-1}{R}^{-1}]}^{-1}$ to get

$$\displaystyle{{({R}^{-1} - {r}^{-1}I)}^{-1} = {({R}^{t} - {r}^{-1})}^{-1} = rRA{(r - a)}^{-k} + \cdots \,.}$$

Now take the transpose of this equation to get

$$\displaystyle{{(R - {r}^{-1})}^{-1} = r{A}^{t}{R}^{t}{(r - a)}^{-k} + \cdots \,.}$$

So far, the assumption that r ∈ Γ has not been used, but now the complex conjugate of the above equation gives, since on Γ, ${r}^{-1} =\bar{ r}$ and ${(R - {r}^{-1})}^{-1} = -({r}^{-1}I - R) = -(\bar{r}I - R)$,

$$\displaystyle\begin{array}{rcl}{ (rI - R)}^{-1}& =& -{r}^{-1}\bar{{A}}^{t}{R}^{t}{({r}^{-1} - {a}^{-1})}^{-k} + \cdots \\ & =& -{r}^{-1}\bar{{A}}^{t}{R}^{t}{r}^{k}{r}^{-k}{({r}^{-1} - {a}^{-1})}^{-k}{a}^{-k}{a}^{k} + \cdots \\ & =& {(-1)}^{k-1}{r}^{k-1}{a}^{k}\bar{{A}}^{t}{R}^{t}{(r - a)}^{-k} + \cdots \,. {}\end{array}$$

(7.29)

Now expand $f(r) = {r}^{k-1}$ in a power series about r = a to get ${r}^{k-1} = {a}^{k-1} + (k - 1){a}^{k-2}(r - a) + \cdots $ . If this expansion is substituted in (7.29), the result is (7.28), and Frobenius’ proof is completed.

See in this connection [267, pp. 245–248].

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Titel: The Cayley–Hermite Problem and Matrix Algebra
verfasst von: Thomas Hawkins
Verlag: Springer New York
Buch: The Mathematics of Frobenius in Context
Print ISBN: 978-1-4614-6332-0

Electronic ISBN: 978-1-4614-6333-7

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-6333-7_7

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