数理解析研究所講究録
第 1996 巻 2016 年 83-91

83

Some contractions and the Poncelet property of
their numerical ranges
Hiroshi Nakazato (Graduate School of Science and Technology,
Hirosaki University, nakahr@hirosaki-u.ac.jp)

1. A special class of contractions
In 1814, a French mathematician Jean-Victor Poncelet [26] described his
famous closure theorem: Let and $D$ be two conics on the complex projective plane. If there exists a closed -gon inscribed in $D$ and circumscribed to$C$ then,stal.ting at an arbitrary point of $D$ , there is a closed -gon inscribed$D$inand circumscribed to (cf. [13]). A rigid proof of Poncelet’s closure
theorem was given by Jacobi [20] based on the elliptic function theory (cf.
[27]). For a pair of two conics
and $D$ on the plane, a point $P\in C$ anda point $Q\in D$ have a relation $P\sim Q$ if there is a tangent line of at $P$
passing through . By this relation the space curve
$C$

$n$

$n$

$C$

$C$

$C$

$Q$

$L=\{(P, Q)\in C\cross D:P\sim Q\}$

has a parametrization by elliptic functions with common modular invariants(cf. [4]). In this sense, is an elliptic curve. From a matrix theoretic view
point, the Poncelet property arises in the boundary of the numerical rangeof some contraction matrices. Letmatrix. The numericalbe anrange of A is defined as
$L$

$A$

$n\cross n$$W(A)=\{\langle Ax, x\rangle:x\in \mathbb{C}^{n}, ||x||=1\}$

and the rank-k numerical range ofset$\Lambda_{k}(A)=${$\lambda\in \mathbb{C}$$A$,(1.1)

is introduced and defined in [7] as the: $PAP=\lambda P$ for some rank$k$orthogonal projection $P$ },. In the case $k=1,$
reduces to $W(A)$ . The rank-k
if $3k\leq n+2$
is a compact convex set andnumerical range$1\leq k\leq n$$\Lambda_{k}(A)$$\Lambda_{k}(A)$$\Lambda_{k}(A)\neq\emptyset$

84(cf. [3, 7, 8, 22 If is a contraction, i.e., $||Ax||\leq||x||$ for any
then its numerical range $W(A)$ is contained in the closed unit disc. Mirmanofmatrices for which the boundary[23] found an important class$C=\partial W(A)$ of the numerical range of a matrix $A\in S_{n}$ and the unit circle$D=\{z\in \mathbb{C} : |z|=1\}$ form a Poncelet pair. Gau-Wu [15] independently
class. For a survey on numericalfound the Poncelet property for therange and the Poncelet property, see for instance [17], and for recent works
on the Poncelet property, see [10, 16, 24]. A formulation of the Poncelet
property of a matrix $A\in S_{n}$ using the complex algebraic geometry was
if is a contraction, has nomatrix is ingiven in [2, 25]. An$(I_{n}-A^{*}A)=1.$
eigenvalue with modulus 1, and rank$A$$x\in \mathbb{C}^{n},$$n\cross n$$S_{n}$$S_{n}$$A$$n\cross n$$S_{n}$$A$$A$

FigurelIn Figure 1, we provide an example of the boundary of the numerical
range of a matrix in . We present two quadrilaterals inscribed in the.
unit circle and circumscribed tomatrices.
The following result characterizes the class of$A$$S_{3}$$\partial W(A)$$S_{n}$. Then there exists
Proposition l.l[Mirman; Gau,P. Y. Wu] Let
an $n\cross n$ unitary matrix so that $UA_{0}U^{*}=(a_{ij})$ is an upper triangular$A_{0}\in S_{n}$

$U$

85
matrix given by(1.2)$a_{\dot{z}j}=\{\begin{array}{ll}a_{\hat{J}}, if i=j;(1-|a_{i}|^{2})^{1/2}(1-|a_{j}|^{2})^{1/2}, if i=j-1;\prod_{k=i+1}^{j-1}(-\overline{ap})(1-|a_{i}|^{2})^{1/2}(1-|a_{j}|^{2})^{1/2}, if i<j-1;0, if i>j;\end{array}$for some $|a_{j}|<1,$ $j=1$ , 2, .. . , $n.$
The numerical range of a matrix $A\in S_{n}$ can be expressed as$W(A)=\cap$ $W(U)$ : $U$ is an $(n+1)$ -dimensional unitary dilation of $A${}(cf. [15, 23]) which also gives a partial answer to Halmos’ conjecture, namely,closure $(W(T))=\cap$ { $clos\alpha re(W(U))$ :$U$

is a unitary dilation of },$T$

for a contraction operator on a complex Hilbert space (cf. [1]). A generalanswer is given by Choi and Li ]. Moreover, it is shown in [14, Theorem
with rank $(I_{n}-A^{*}A)=k$ has a general
1.2] that ancontraction
consequence:
$T$

$|9$

$A$

$n\cross n$$\Lambda_{k}(A)=\cap${ $W(U)$ :$U$

is an $(n+k)$ -dimensional unitary dilation of $A$ }.

matrix (1.2). Then any
Proposition 1.2 [Gau, Wu]. Let $A=(a_{ij})$ be a$(n+1)\cross(n+1)$ unitary dilation ofis unitarily equivalent to a member
given byof a one-parameter family of unitary matrices$S_{n}$$A$$B(\lambda)=(b_{ij}(\lambda))$$b_{ij}(\lambda)=\{\begin{array}{ll}a_{ij}, if 1\leq i,j\leq n;\lambda(1-|a_{j}|^{2})^{1/2}, if i=n+1,j=1;\lambda(\prod_{k=1}^{j-\lambda}(-\overline{a_{k}}))(1-|a_{j}|^{2})^{1/2}, if i=n+1, 2\leq j\leq n;(1-|a_{i}|^{2})^{1/2}, if j=n+1, i=n;(\prod_{k=;+1}^{n}(-\overline{a_{k}}))(1-|a_{i}|^{2})^{1/2}, if j=n+1, 1\leq i\leq n-1;\lambda\prod_{k=1}^{n}(-\overline{a_{k}}) , if i=j=n+1;\end{array}$(1.3)

where$\lambda$

is a parameter on the unit circle $|z|=1.$

86

2. $The$ algorithm generating new Poncelet pairs
In [2], a complex algebraic formulation was given for $A\in S_{n}$ . In [6], newPoncelet pairs are found. Let be amatrix (1.2) and
its unitarydilation matrix (1.3). We present an algorithm that computes the defining
polynomial $L(X, Y)$ which produces a new part: $L(X, Y)=0$ of the
new Poncelet curve with respect to the boundary generating curve of $W(A)$ .
$A$$S_{n}$$B(\lambda)$$C_{P}$Algorithm

Figure 2; new Poncelet pair$\bullet$$\bullet$

Step 1 Computeform (1.3).

$F_{B(\lambda)}(t, x, y)$associated with the matrix

Step 2 Substitute $y=-1/Y-xX/Y$ into
pQlynomial$H(x, X, Y : \lambda)$$=$$F_{B(\lambda)}(t, x, y)$of theand define a$Y^{n+1}F_{B(\lambda)}(1, x, -1/Y-xX/Y)=F_{B(\lambda)}(Y, xY, -1-xX)$$= c_{n+1}(X, Y)x^{n(n+1)}+\cdots+c_{0}(X, Y)$$\bullet$$B(\lambda)$.

Step 3 Take the resultant $R(X, Y : \lambda)$ of $H(x, X, Y : \lambda)$ and $H_{x}(x,$ $X_{\}}Y$ :with respect to $x.$$\lambda)$$\bullet$

Step 4 Find a factor polynomial $K(X, Y : \lambda)$ of the resultant $R(X,$of total degree $(n+1)n/2$ in $X,$ $Y$ with multiplicity 2.

$\lambda)$

$Y$:

87
$\bullet$$\bullet$$\bullet$

Step 5 Substitute $\lambda=((1-s^{2})+2is)/(1+s^{2})$ into $K(X, Y;\lambda)$ and$K_{X}(X, Y;\lambda)$ .
Step 6 Take the respective numerators $\tilde{K}(X, Y;s)$ and $\tilde{K}_{X}(X, Y;s)$of $K(X, Y;s)$ and $K_{X}(X, Y;s)$ .

Step 7 Compute the Sylvester’s resultant $S(X, Y)$ of $\tilde{K}(X, Y;s)$ andwith respect to$\tilde{K}_{X}(X, Y;s)$$\bullet$$s.$

Step 8 Find a factor $L(X, Y)$ of $S(X, Y)$ with multiplicity 2.

In Figure, we present the graphic of a new Poncelet curve for a matrix
in . The union of the curve labeled 4 and the curve labeled 2 is $L(X_{\}}Y)=$.
. The curve labeled 3 is
O. The curve labeled 1 is$A$

$S_{4}$$\partial W(A)$$\partial\Lambda_{2}(A)$

Example. Let $n=3$ and$B(\lambda)=(0a0$$-\lambda a\sqrt{1-a^{2}}1-a^{2}a0$$\lambda a^{2\sqrt{1-a^{2}}}-a\sqrt{1-a^{2}}1-a^{2}a$$-a^{2\sqrt{1-a^{2}}}a-\lambda a^{3}$for is a positive real number less than 1. Then the polynomial $L(X, Y)$which gives the equation $L(X, Y)=0$ of the new Poncelet curve is given by$a$$L(X, Y)=6a(-a^{2}+1)XY^{2}+(a^{6}+3a^{2}-4)Y^{2}+2a(a^{2}+3)X^{3}$$+(a^{6}-21a^{2}-4)X^{2}+6a(-a^{4}+3a^{2}+2)X+(a^{6}-9a^{2})$.

3. Matrices unitarily similar to complex symmetricmatrices
In this section we present a result related with the inverse problem for theshape of a numerical range (cf. [18]).

Theorem 3.1(cf.[5]). Every matrix inplex symmetric matrix.

$S_{n}$

is unitarily similar to a com-. Then by [16,. Corollary 1.3] (see also [23] [Theorem4 the matrix
has a canonical upper triangular form. The matrix
also dilates to an $(n+1)\cross(n+1)$ unitary matrix $W$ with distinct eigenvalues (cf. [15, Lemma 2.2]). We assume the distinct eigenvalues of $W$ are

Proof. Let$A\in 8_{n}$
$A$

$A$

88
given by,, and their respective corresponding eigenvectors are, ,. Let be the -dimensional orthogonal projection satisfying$A=(PWP)|_{C^{n}}$ . By replacingbyfor some angles , . .. ,the space $C^{n}=P(C^{n+1})$ is expressed as$f_{1},$$c_{1},$$f_{2}$$\cdots$$c_{2}$$\cdots$,$c_{n+1}$$P$$f_{n+1}$$n$$f_{j}$$\exp(i\theta_{j})f_{j}$$\theta_{n+1},$$\theta_{1}$$C^{n} = \{z_{1}f_{1}+z_{2}f_{2}+\cdots+z_{n+1}f_{n+1}:(z_{1}, \ldots, z_{n+1})\in C^{n+1},$$b_{1}z_{1}+b_{2}z_{2}+\cdots+b_{n+1}z_{n+1}=0\}$

for some non-negative real numbers. Since the modulus of
any eigenvalue of is strictly less than 1, the numbers are positive. Thenthe space $C^{n}=P(C^{n+1})$ consists of the linear spans of$b_{1},$$b_{2},$$\rangle b_{n+1}$$A$$b_{j}$$\{b_{1}f_{2}-b_{2}f_{1}, b_{1}f_{3}-b_{3}f_{1}, .

.

.

, b_{1}f_{n+1}-b_{n+1}f_{1}\}$.

(3.1)

Let
be an orthonormal basis of $C^{n}=P(C^{n+1})$ obtained by
the Gram-Schmidt orthonormalization of independent vectors in (3.1).
The vectors are expressed as$\{\xi_{1}, \xi_{2}, \cdots, \xi_{n}\}$$n$$\xi_{j}$$\xi_{j}=\xi_{j,1}fi+\xi_{j,2}f_{2}+\cdots+\xi_{j,n+1}f_{n+1}$

for some real numberswith $\xi_{j,j+1}>0$ and $\xi_{j,j+2}=\xi_{j,j+3}=\cdots=0\dot{ノ},$$j=1$ , 2, . . . , . With respect to the orthonormal basis, theoperator on the -dimensional Hilbert spacesatisfies the property$\xi_{j,k}$

$n$

$A$

$\{\xi_{1}, .

.

.

, \xi_{n}\}$$C^{n}$$n$$\langle A\xi_{\ell}, \xi_{k}\rangle=\sum_{j=1}^{n+1}c_{j}\xi_{\ell,j}\xi_{k,j}=\sum_{j=1}^{n+1}c_{j}\xi_{k,j}\xi_{l,j}=\langle A\xi_{k},\xi_{l}\rangle.$

Thus the operator has a symmetric matrix representation with respect tothis orthonormal basis$A$$\{\xi_{1}, .

.

.

, \xi_{n}\}.$$\square$

We are interested in matrices unitarily similar to complex symmetricmatrices. In [18] Helton and Spitkovsky proved that every
complexmatrix has ancomplex symmetric matrix
satisfying $W(A)=$$W(B)$ . This result follows from the followin theorem.
$n\cross n$

$A$

$B$

$n\cross n$

Theorem 3.2(Helton and Vinnikov [19]). Suppose that $F(x, y, z)$ is adegree ternary homogeneous polynomial with real coeﬃcients for which
the equation $F(\cos\theta, \sin\theta, z)=0$ in has real solutions for every angle$0\leq\theta\leq 2\pi$ and $F(O, 0,1)=1$ .
Then there exist $n\cross n$ real symmetric$H$$G,$matrices
satisfyng
$n$

$z$

$n$

$F(x, y, z)=\det(xH+yG+zI_{n})$ .

89This result prove\‘a that the conjecture posed by P. Lax [21](page 184) is true.
In [11], page 95, M. Fiedler posed a similar conjecture by relaxing $H,$ byHermitian matrices. In $|12$ ], Fiedler proved the assertion of Theorem 3.2 inthe case $F(x, y, z)=0$ is a rational curve.
$G$

In [28] T. Takagi proved that every Toeplitz matrix is unitarily symmetricto a complex symmetric matrix.

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