数理解析研究所講究録903 巻 1995 年 30-38

30

the genertalized Numerical Range and its Boundary
Hiroshi Nakazato (Faculty of Science, Hirosaki Univ. ,related top
of $C$ -numeric range of a mat $rix$ andsi der the boundAbstract. Weof in varian polynomials$\dashv*\neg\backslash \vee\iota$$i_{arrow}^{-}\underline{{}^{\mathrm{t}}s^{*1}-}$$\backslash \backslash 1$$co\mathrm{n}$$al$$a\mathrm{r}y$$\mathrm{i}ts$$\mathrm{i}cs$

in the theory

$t$

range

generalized numerical

the

1.

We recall the definition of thematrices.
Def. Let $A,$ $C$ be$n\cross n$$C$matrix asof a-numerical range. Denote bysemi-algebraica$M_{n}(\mathrm{C})$set.

the set of all $n\cross n$ complex

complex matrices. Set$W_{C}(A)=\{tr(CUAU*) :(1.1)

U\in U(n)\}$act group of all unit $ary$ matrices of oder . $W_{C}(A)$ is alled the -numerical range
where $U(n)$ is the.
of
It is clear that $C$-numerical range is a unitary invariant of the suare matrix : If $B$ is unitarily similar to$B=U$ A, then $W_{C}(B)=W_{C}(A)$ . If $C$ is a rank one orthogonal projection, then the C-numerical
range $W_{C}(A)$ coincides with the classical numerical range $W(A)$ ,
$C$

$c$

$n$

$co\mathrm{m}p$

$A$

$A$

$U^{*}$$A,\mathrm{i}.\mathrm{e},$$W(A)=\{(A\xi, \xi) :\xi\in \mathrm{C}^{\mathrm{n}}, ||\xi||=1\}$.

The generalized numerical range $W_{C}(A)$ is the range of the real algebraic variety$U(n)= \{U=\{u_{i,j}\}_{1\leq i,j\leq n}\in \mathrm{C}^{\mathrm{n}^{2}}\cong \mathrm{R}^{2\mathrm{n}^{2}} :\sum_{\mathrm{k}=1}^{\mathrm{n}}\mathrm{u}\mathrm{i},\mathrm{k}\overline{\mathrm{u}\mathrm{i},\mathrm{k}}=\delta_{\mathrm{i},\mathrm{j}}(1\leq \mathrm{i},\mathrm{j}\leq \mathrm{n})\}$(1.2)

under the polynomial mapping(1.3).

$X(\in M_{n}(\mathrm{C})=\mathrm{C}^{\mathrm{n}^{2}}\cong \mathrm{R}^{2\mathrm{n}^{2}})-\succ tr(CxAx^{*})(\in \mathrm{C}\cong \mathrm{R}2)$

We recall Tarski
Def. A bset A of theBoolean algebra gen era

theorem.

$=\mathrm{S}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}’ \mathrm{s}$$s\mathrm{u}$en sional affine spaceby the family of sets$n-d\mathrm{i}\mathrm{m}$$t\mathrm{e}\mathrm{d}$$\mathrm{R}^{\mathrm{n}}$$Z_{0}(f)=\{x=(x_{1}, X_{2}, \ldots, x_{n})\in \mathrm{R}^{\mathrm{n}} :is aid to be semi-algebraic if A is an element of the$s$\mathrm{f}(\mathrm{x}_{1},\mathrm{x}_{2,..}\nu \mathrm{X})=0\}$,

and$Z_{1}(f)=\{x=(x_{1}, X_{2}, \ldots, x_{n})\in \mathrm{R}^{\mathrm{n}} :where varies over the real polynomial ring
In other words, A :a semi-algebraic set$f$\mathrm{f}(\mathrm{x}_{1},\mathrm{x}_{2}, \ldots , \mathrm{x}_{\mathrm{n}})>0\}$$\mathrm{R}[\mathrm{X}_{1}, \mathrm{x}_{2)}\ldots , \mathrm{X}_{\mathrm{n}}]$$(\subset \mathrm{R}^{\mathrm{n}})$$\Leftrightarrow \mathrm{d}\mathrm{e}t\mathrm{i}\mathrm{n}:\mathrm{t}$$\Lambda=\bigcup_{jk}^{m}=\mathrm{l}=\mathrm{n}^{l_{\mathrm{j}}}1$

where
Of course, an algebraic set.

:$Z_{\epsilon_{\mathrm{j},k}}(f_{j},k)$.

$\epsilon_{j,k}\in\{0,1\},$ $f_{j,k}\in \mathrm{R}[\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{X}_{\mathrm{n}}]$$(\subset \mathrm{R}^{\mathrm{n}})$is semi-algebraic.

theorem.
a Corollary of Tarski
of a semi-algebraic set A
The range, is semi-algebraic, where$=\mathrm{S}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}’ \mathrm{s}$$\subset \mathrm{R}^{\mathrm{m}}$$\phi(\Lambda)$

under a polynomial mapping.

$\phi_{j}\in \mathrm{R}[\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots, \mathrm{X}_{\mathrm{m}}](1\leq \mathrm{j}\leq \mathrm{n})$$\mathrm{R}^{\mathrm{m}}arrow \mathrm{R}^{\mathrm{n}}$A Simple Example$\mathrm{S}^{2}=\{(\mathrm{x}_{1}, \mathrm{x}_{2,3}\mathrm{X})\in \mathrm{R}^{3} :\mathrm{x}_{1}^{2}+\mathrm{x}_{2}^{2}+\mathrm{x}_{3}^{2}=1\}\subset \mathrm{R}^{3}$

1 projection
A closed disc$\{(X_{1}, X_{2})\in \mathrm{R}^{2} :\mathrm{x}_{1}^{2}+\mathrm{x}_{2}^{2}\leq 1\}$.

We remark that the closed disc is semi-algebraic and not algebraic.

$\phi=(\phi_{1}, \phi_{2}, \ldots, \emptyset n)$:

31

Another Result about Semi-Algebraic Sets. The boun daryalso semi-algebraic.

$\partial\Lambda$

of a semi-algebraic set$\Lambda(\subset \mathrm{R}^{\mathrm{n}})$isUsing the result above, Takaguchi,Nishikawa and I proved the following.
of $C$ -numerical range $W_{C}(A)\subset$]] The bounbary
Theorem.
is the union of a finite number of real algebraic arcs.
$\partial W_{C}(A)$$\mathrm{f}T\mathrm{a}k\mathrm{a}g\mathrm{u}c\Lambda \mathrm{j}=Nishik\mathrm{a}wa=N\mathrm{a}kaZ\mathrm{a}tof\mathit{1}$$\mathrm{C}\cong \mathrm{R}^{2}$

Furthermore, using the theorem above and Bezout’s theorem, we proved the following.

Theorem. The number of non
2. Concrete$\mathrm{C}$$\omega$-regu $lar$ points ofis finite.

$\partial W_{C}(A)$-numerical ranges

Second I consider the -numerical ranges of matrices.
We set$q$.

$C=C_{q}=(_{0}^{q}0.\cdot$.

$\sqrt{1-|q|^{2}}00.\cdot$

We denote $Wc_{l}(A)$ by $W(A:q)$ . We recall$\mathrm{N}.\mathrm{K}$.

$0$$0..\cdot.\cdot$$.000.$

.

$00$

)

.

.Tsing’s formula.

A formula due to Tsing.

$W(A:q)=\cup\{q(A\xi,\epsilon)+.\sqrt{1-|q|^{2}}r\mathrm{e}x\mathrm{p}(i\theta)\sqrt{||A\xi||^{2}-|(A\xi,\xi)|2}.:\epsilon\in \mathrm{C}^{\mathrm{n}}, ||\xi||=1,0\leq \mathrm{r}\leq 1,0\leq\theta\leq 2\pi\}$(2.2).

This formula is rewriten as$W(A:q)=\cup\{B(qZ:\sqrt{\phi_{A}(z)} :where$\phi_{A}(z)=\max\{||A\xi||^{2}-|z|^{2} :(2.3),

z\in W(A:1)\}$.

\xi\in \mathrm{C}^{\mathrm{n}}, ||\xi||=1, (\mathrm{A}\xi,\xi)=\mathrm{z}\}$for $z\in W(A)=W(A:1)$ .

(2.5),We set$\psi_{A}(z)=\max\{||A\xi||^{2} :We review

Theorem.
on $W(A)$ .

[$\mathrm{N}.\mathrm{K}$$\mathrm{N}.\mathrm{K}$\xi\in \mathrm{C}^{\mathrm{n}}, ||\xi||=1, (\mathrm{A}\xi,\xi)=\mathrm{z}\}$.

.Tsin’s results.

.Tsing [6]]$\psi_{A}$is a concave function on $W(A)$ and hence$\phi_{A}$and$\sqrt{\phi_{A}}$are also concave

Using the theorem above we can prove the following two claims.

Proposition. $A,$

$B_{1},$$\ldots,B_{m}$: matrices. $W(A)= \bigcup_{j=1}^{m}W(B_{j}),$ $\phi A(Z)=\max\{\phi_{B_{j}}(z): 1\leq j\leq m\}$ ,$\Rightarrow W(A : q)=\bigcup_{j}^{m}W=1(Bj :for every

$q$

q)$

with $|q|\leq 1$ .

Theorem. [2] (1) Suppose that
equation holds for every$]$$\lambda_{1},$$\lambda_{n}$$\lambda_{2},$$\ldots,$

are arbitrary complex numbers $(n\geq 3)$ . Then the following$z\in \mathrm{c}_{\mathrm{o}\mathrm{n}\mathrm{V}}(\{\lambda 1, \lambda 2, \ldots, \lambda_{\mathrm{n}}\})$:$\phi_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(}\lambda_{1},\lambda 2,\ldots,\lambda \mathrm{n})(Z)=\max\{\phi_{\mathrm{d}}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda_{j_{1}},\lambda \mathrm{j}2’\lambda_{\mathrm{j}_{3}})(Z):1\leq j1<j_{2}<j3\leq n\}$

.

Hence
$W(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda_{1}, \lambda 2, \ldots, \lambda \mathrm{n}):\mathrm{q})=\cup\{\mathrm{W}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda \mathrm{j}_{1}, \lambda \mathrm{i}2’ s)\lambda_{\mathrm{j}}:\mathrm{q}):1\leq \mathrm{j}_{1}<\mathrm{j}_{2}<\mathrm{j}_{3}\leq \mathrm{n}\}$(2) If$\lambda_{1},$$\lambda_{2},$$\partial W$$\lambda_{3}\in \mathrm{C}$(diag$(\lambda 1,$

and$\lambda 2,$$|\lambda_{1}|=|\lambda_{2}|=|\lambda_{3}|=$$\lambda_{3})$: )$\mathrm{q}$$\subseteq\{\mathrm{z}\in \mathrm{C} :.

l,then|\mathrm{z}|=1\}\cup${$\partial \mathrm{W}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\lambda \mathrm{i},$$\lambda \mathrm{k})$: q) :$1\leq \mathrm{j}<\mathrm{k}\leq 3$

}.

32

3. -numerical ranges of 2 by 2 matrices
The study of $C$ -numerical ranges ofmatrices is reduces to that of$\mathrm{C}$$2\cross 2$$W(:q)$
for$0\leq\beta\leq\alpha,$$0\leq q\leq 1$.

We have the following:$\mathrm{f}_{\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}}1\mathrm{a}(\mathrm{C}\mathrm{f}.[3])$$W(:q)$$=\{r[(\alpha+\beta)/2+\sqrt{1-q^{2}}(\alpha-\beta)/2]\cos(\theta)+\sqrt{-1}\mathrm{r}[(\alpha-\beta)/2+\sqrt{1-\mathrm{q}^{2}}(\alpha+\beta)/2]\sin(\theta)$$0\leq\theta\leq 2\pi\},where0\leq\beta\leq\alpha,$ $0\leq q\leq 1$.

:$0\leq \mathrm{r}\leq 1$,

4. Relation between generalized numerical ranges and invariant polynomials
It is well known that the following three conditions formatrices $x,$ are mutually equivalent: (i) $W(x)=$tr(x)
for some unitary matrix.
$2\cross 2$$y$$=\mathrm{t}\mathrm{r}(\mathrm{y}),\mathrm{t}\mathrm{r}(\mathrm{x}^{2})=tr(y^{2}),ir(xX)*=tr(y^{*}y);(iii)y=UxU^{*}$$W(y);(\mathrm{i}\mathrm{i})$Formatricesthe following three conditions are mutually equivalent: (iv) $W_{P}(x)=W_{P}(y)$$\mathrm{W}_{H}(x)=W_{H}(y)$ for every Hermitian matrix
for every orthogonal projection; (vi)
for every.
$n\cross n$$x,$ $y$$P;(\mathrm{v})$$\mathrm{H}$$(\alpha \mathrm{x}+\beta \mathrm{x}^{*}))=\det(\lambda \mathrm{I}\mathrm{n}-(\alpha \mathrm{y}+\beta \mathrm{y}^{*}))$$\lambda,$

In the case $n=3$ , the condition (iv) is equivalent to the condition: $W(x)=W(y)$ , tr(x)Question.

$x,$ $y:3\cross 3$$\mathrm{d}\mathrm{e}t(\lambda \mathrm{I}_{\mathrm{n}}-$$\alpha,\beta\in \mathrm{C}$$=\mathrm{t}\mathrm{r}(\mathrm{y})$.

matrices such thattr(x)$=\mathrm{t}\mathrm{r}(\mathrm{y}),$$\mathrm{w}(\mathrm{X}:\mathrm{q})=\mathrm{W}(\mathrm{y}:\mathrm{q})$for every $0\leq q\leq 1$ .
$\Rightarrow$?

$\exists_{U}$: a unitary matrix such that$\mathrm{y}=\mathrm{U}\mathrm{x}\mathrm{U}^{*}\mathrm{o}\mathrm{r}\mathrm{y}=\mathrm{U}(^{\mathrm{t}_{\mathrm{X}}})\mathrm{U}^{*}$.

I had the question above under the influence of the following theorem.

Theorem. (S.Teranishi,Nagoya Math.J. $1986[8]$ ). Let$x,$ $y$be arbi trary$3\cross 3$

complex matrices. (1) If(4.1),$\mathrm{t}\mathrm{r}(\mathrm{X}^{\mathrm{k}})=\mathrm{t}\mathrm{r}(\mathrm{y}^{\mathrm{k}})(\mathrm{k}=1,2,3)$tr(x*x)then there exits a unitary matrix$U$$\mathrm{t}\mathrm{r}(\mathrm{x}^{*2}\mathrm{X})=\mathrm{t}\mathrm{r}(\mathrm{y}^{*2}\mathrm{y})$(4.3),$\mathrm{t}\mathrm{r}(\mathrm{x}^{*}\mathrm{x})22=\mathrm{t}\mathrm{r}(\mathrm{y}^{*2}\mathrm{y}^{2})$(4.4),

such that$y=UxU^{*}$. (2) If the conditions(4.2),$=\mathrm{t}\mathrm{r}(\mathrm{y}\mathrm{y})*$$(\mathit{4}.\mathit{1}),(\mathit{4}.2),(\mathit{4}.\mathit{3}),(\mathit{4}.\mathit{4})$or$\mathrm{y}=\mathrm{U}(^{\mathrm{t}}\mathrm{x})\mathrm{u}*$holds and if(4.5),$\mathrm{t}\mathrm{r}(\mathrm{x}^{*2}\mathrm{X}^{2_{\mathrm{X}^{*}}}\mathrm{x})=\mathrm{t}\mathrm{r}(\mathrm{y}^{*2}\mathrm{y}^{2}\mathrm{y}^{*}\mathrm{y})$then there exits a unitary matrix

We denote by

the

$U$

such that $y=UxU^{*}$ .

-algebra of allcomplex matrices. I want to find a finite subset $F$ ofsuch that $\{W_{C}(\cdot) : C\in F\}$ is a complete unitary invariant system. This desire is
influenced by the following theorem.
$M_{n}(\mathrm{C})$$M_{n}(\mathrm{C})$$C^{*}$$n\cross n$$(\mathrm{c}\mathrm{f}.[7],\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}10)$

33

TheoremLet$(\mathrm{c}\mathrm{f}.[9])$$x,$ $y$becomplex matrices. Set $x_{1}=x,$ $x_{2}=x^{*},$ $y_{1}=y,$ $y_{2}=y^{*}$ . If$n\cross n$$\mathrm{t}\mathrm{r}(\mathrm{x}_{\mathrm{i}_{1}}\mathrm{x}_{\mathrm{i}_{2}}\cdots \mathrm{x}_{\mathrm{i}_{\mathrm{p}}})=\mathrm{t}\mathrm{r}(\mathrm{y}:1\mathrm{y}:_{2}\cdots x_{\mathrm{i}_{\mathrm{p}}})$

for every $l\leq p\leq 2^{n}-1,1\leq i_{1},$$i_{2},$$\ldots,$$i_{p}\leq 2$, then there exits a unitary matrix$U$such that $y=UxU^{*}$ .

We can obtain another criterion by using the following lemma.
such that $I_{n}\in A,$ $I_{n}\in B.$ If is a -isomorphism-subalgebras ofLemma. Let $A,$ be$x\in
A$ , then there exits a unitary matrix
such th atevery
tr(x) forof onto satisfying$\phi(x)=UxU^{*}$ for every $x\in A$ .
$B$$C^{*}$$B$$A$$\phi$$M_{n}(\mathrm{C})$

$*$

$U$

$\mathrm{t}\mathrm{r}(\phi(\mathrm{x}))=$

Proposition. If...

$\mathrm{t}\mathrm{r}(\mathrm{x}_{\mathrm{i}_{1}}\mathrm{x}_{\mathrm{i}_{2}}$for every $1\leq p\leq 4(n^{2}-3),$$1\leq i_{1},$ $i_{2},$$\ldots,$$i_{\mathrm{p}}\leq 2$$\mathrm{x}_{\mathrm{i}_{\mathrm{p}}})=\mathrm{t}\mathrm{r}(\mathrm{y}:_{1}\mathrm{y}_{\mathrm{i}_{2}} ...

, then th$\mathrm{e}re$\mathrm{y}_{\mathrm{i}_{\mathrm{p}}})$

exists a unitary ma trix , such th at $y=UxU^{*}$ .
$U$

This proposition is stronger than the theorem above under the condition$n\geq 8$.

We have an analogous reslut.
Proposition. If...

$\mathrm{t}\mathrm{r}(\mathrm{x}_{\mathrm{i}_{1}}\mathrm{x}_{\mathrm{i}_{2}}$for every $1\leq q\leq 2(n^{2}-3),$
preserving ear bijection of the.. .

$\mathrm{y}_{\mathrm{i}_{\mathrm{q}}})$, then there exists a -preserving, Hilbert-Schmidt norm.
onto the -algebra$*$$1\leq i_{1},$ $i_{2},$$li\mathrm{n}$$x_{\mathrm{i}_{\mathrm{q}}})=\mathrm{t}\mathrm{r}(\mathrm{y}_{\mathrm{i}_{1}}\mathrm{y}_{\mathrm{i}_{2}}$$C^{*}$$\ldots,$$i_{q}\leq 2$-algebra $C^{*}(x)$$C^{*}$$C^{*}(y)$5. 3 by 3 orthostochastic matrices

We consider a$3\cross 3$

real matrix$A(x,y, z:u, v, w)==xe_{1}+ye_{2}+ze_{3}+ue_{4}+ve_{5}+we_{6}$ ,for $x\geq 0,$ $y\geq 0,z\geq 0,$ $u\geq 0,v\geq 0,w\geq 0,$ $x+y+z+u+v+w=1$ . Here$e_{1}=,e_{2}=$ , $e_{3}=$ .
$e_{4}=$ , $e_{5}=$ , $e_{6}=$ .
doubly stochastic matrix. A $(3\cross 3)$ doublystochastic
Such a matrix $A(x,y, z : u, v, w)$ is called amatrix is said to be orthostochastic if it is the Hadamard product (entrywise product) of some $(3\cross 3)$. The necessar.y and sufficient condition of Au-Yeng and Poon
unitary matix and its conjugate
for a doubly stochastic matix$3\cross 3$

$D$

$U$

$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}_{\mathrm{X}}\overline{U}$$A(x,y, z : u, v,w)$

to be orthostochastic$(\mathrm{c}\mathrm{f}.[5])$is stated in anther fashion$(\mathrm{C}\mathrm{f}.[4])$:$x^{2}y^{2}+x^{2}z^{2}+y^{2}z^{2}-2xyz(X+y+z)+u^{2}v^{2}+u^{2}w^{2}+v^{2}w^{2}-2uvw(u+v+w)$$-4uvw(x+y+z)-4xyZ(u+v+w)-2(xy+xz+yz)(uv+uw+vw)\leq 0$
Using this characterization we can$\mathrm{o}\mathrm{b}t$

ain the following proposition.

.

(5.1)

34

Proposition ([4]) Suppose thatis the affine hull of the compact convex set of allis the compact set of all
matrices andorthostochastic matrices. Denote byin theace . Then the following
tion holds:$3\cross 3$$A_{3}$$3\cross 3$$O_{3}$$O_{3}$$A_{3}$$a\mathrm{f}\mathrm{f}\mathrm{i}_{I1}eS\mathrm{P}$$\partial O_{3}$doubly stochastic
the boundary of$eq\mathrm{u}a$$\partial O_{3}=\{g\circ g :g\in SO(3)\}$,where $SO(3)$ is the group of rota tions in the 3-dimensional Euclidean space.

Consider a linear functional$\Psi$of the complex vector space$\Psi$$M_{3}(\mathrm{C})$,: $x=\{X_{i,j} : 1\leq i,j\leq 3\}\in M_{3}(\mathrm{c})rightarrow(x_{11}+X12+x13+X_{21}+x_{22}+X23+x_{31}+X_{3}2+x33)$ .

Suppose that

$A$

and

$C$

are complex$3\cross 3$

diagonal matrices with diagonal entriries$a=\{a_{1}, O2, a_{3}\},$ $c=\{C_{1},C_{2}C3\})$

respectively. Consider the$3\cross 3$

matrix $[A:C]$ given as the product$[A:c]=$
Then the$C$-numerical range of$A$$(^{\ell}a)c,\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$is,.

coincides with the set$\{\Psi([A : c]\circ c) :G\in O_{3}\}$.

The following proposition is effective to determine the $C$ -numerical range of a matrix$A=C$ andis acomplex diagonal matrix.
$A$

$A$

under the condition$3\cross 3$

Proposition ([4]) Suppose that ia a complex
range $W_{A}(A)$ coincides with the set$A${$\Psi([A:A]\mathrm{o}G)$$3\cross 3$

complex diagonal matrix. Then the A-n umerical: $G\in O_{3}$ and$(^{\mathrm{t}}\mathrm{G})=\mathrm{G}$}.

Under the condition$A=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{1,0, \alpha\}$,the description of the boundary of the range $W_{A}(A)$ concerns deeply the curve ” (Steiner’s) Deltoid”. We
define the Deltoid $D[z_{0} : K : \exp[i\eta]]$ ( is a complex number,If is a positive number, is a real number)in the complex plane $C$ by the following : Choose( is a real number) ia an arbitratry cubic root
of $\exp[i\eta]$$z_{0}$$\eta$$\exp[i\theta]$

.

$D$

[

$z_{0}$$\theta$: If : $\exp[i\eta]$ ] $=\{z_{0}+(K/3)\exp[i\theta](2\exp[2i\phi]+\exp[-i\phi]) : 0\leq\phi\leq 2\pi\}$ .

We obtain the following theorem.
Theorem ([4]) Suppose thatand $\alpha=a+ib$ where
the boundary of the range $W_{A}(A)$ is a subset of the following set$A=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{1,0, \alpha\}$$\{(1-t)(1+\alpha^{2})+t :0\leq t\leq 1\}\cup\{(1-t)(1+\alpha^{2})+2t\alpha :$\cup D1^{z}0$

$a,$

$b$

are real numbers with $b\neq 0$ . Then0\leq t\leq 1\}\cup\{(1-t)(1+\alpha^{2})+t(\alpha)^{2} :$Ic:\exp[i\eta]]$where$z_{0}=(1/(4b^{2}))\alpha(\overline{\alpha}-1)(3a-23a-b2+i[b+4ab])$$I\mathrm{f}=(3/(4b^{2}))|\alpha-1|^{2}|\alpha|^{2}$,,0\leq t\leq 1\}$35.

$|\alpha|^{2}|\alpha-1|^{2}\exp[i\eta]=-(a-a+b-2ab2+b^{2})(a-a-2b+2ab+b^{2})+i(-2b)(2a-1)(a-a-2b^{2})$We give you some graphics about$C$-numerical ranges after this document.

References[1] H.Nakazato,Y.Nishikawa and M.Takaguchi:on the boundary of the -numerical range of a matrix,$\mathrm{c}$preprint 1994 to appear in ”LAMA”[2] H.Nakazato :the Boundary of the Range of a Constrained Sesquilinear Form, preprint 1994,

submitted to LAMA”-numerical range of a 2 x2 matrix, Sci.Report Hirosaki Univ,[3]
in preparation[4] H.Nakazato: Set of 3 x3 orthostochastic’)$\mathrm{H}.\mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{Z}\mathrm{a}\mathrm{t}_{0}:\iota \mathrm{h}\mathrm{e}\mathrm{C}$$41(1994),\mathrm{P}\mathrm{P}$197-206$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{S}_{)}$[5] Y.H.Au-Yeng and Y.T.Poon :$3\cross 3$

orthostochastic matrices and the convexity of the generalized

numerical ranges, Lin.alg.appl. 27(1979) pp 69-79[6] N.K.Tsing : The constrained bilinear form and the -numerical range, Lin.alg.appl.56 (1984)$\mathrm{C}$

pp195-206[7] M.Goldberg and E.G.

$\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{u}\mathrm{S}\mathrm{s}:\mathrm{E}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\iota \mathrm{a}\mathrm{r}\mathrm{y}$Inclusion Relations for Generalized Numerical Ranges,

Lin.alg.appl.18(1977) pp 1-24
[8] Y.Teranishi:the Ring of Invariants of Matrices, Nagoya Math.Journ.104(1986) pp149-161
matrices, Adv.in Math. 19(1976) pp 306-381[9] C.Procesi :The invariant eory of$\mathrm{t}\tilde{\mathrm{h}}$$\mathrm{W}_{C}$(I I-3)$n\cross n$[C)

C– di ag { 1 , 0 , 2$+3$i }.

36Graphics of3$\mathrm{C}$-numerical Ranges of Matrices and the set ofsymmetric orthostochastic matrices$\mathrm{x}3$(I)The$\mathrm{q}$-numerical range of a 3

entries of A are(i) The diagonalroots of(ii)

cOmplex diagonal matrix A

mutually distinct cubic

1.

$\mathrm{q}--4/5$(I I) he$\mathrm{C}$$\mathrm{T}$mat$\mathrm{r}\mathrm{i}\mathrm{x}$(II-1)-numer cal rang$\mathrm{i}$clo seof th$\mathrm{e}3\mathrm{x}3\mathrm{c}\mathrm{o}$mPle$\mathrm{x}\mathrm{d}\mathrm{i}$ago nalare mutually disdistinct cubicth thesed nu me cal rang co

The diagonal entries of$\mathrm{d}\mathrm{d}\mathrm{o}\mathrm{m}$$\mathrm{x}=2\mathrm{c}\mathrm{o}\mathrm{s}$$\mathrm{e}1i\mathrm{c}(\mathrm{C})$.

roo ts of 1. Thi$(\mathrm{I}-|)$$\mathrm{x}3$a$\mathrm{i}\mathrm{n}\mathrm{s}$(2 t)$\mathrm{s}$generalurounde$+\mathrm{c}\mathrm{o}\mathrm{s}(2\mathrm{t}),$$\mathrm{d}$$\mathrm{i}\mathrm{z}$$\mathrm{C}$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}$$\mathrm{r}\mathrm{i}$$\mathrm{e}$by th$\mathrm{y}--2\mathrm{s}$$\mathrm{e}$De lto idin (2t)$\mathrm{D}(0,3,1)$$-\mathrm{s}\mathrm{i}\mathrm{n}(2\mathrm{t})$:$\mathrm{w}\mathrm{i}$$\mathrm{e}$37(I I-2).

On$\mathrm{C}--\mathrm{d}\mathrm{i}$$\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}$Phag$\mathrm{i}\mathrm{c}$

$\{$

de

1 ,

$0$

$\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{i}$be$\mathrm{p}_{\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{p}\mathrm{l}_{0}\mathrm{t}3\mathrm{D}\backslash \backslash$

(1/4)

,
$\mathrm{s}$$+\mathrm{i}$(1/2)

the range$1i\mathrm{c}(\mathrm{C})\mathrm{b}$900)$\mathrm{e}$Plane).

’as ed on the comman

inOther thre

its boundary based on the command

on th

.

(2-parameter Plot)

hardware : APPle , Quadra$\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{t}$$\}$$\sim$$\mathrm{e}$Nathematica$\backslash \backslash$$\mathrm{d}$(ver2.0,

graPhics descri be$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{t}\backslash \backslash$(l-parameter38(III) Theset of 3$\mathrm{x}3$symmetric orthostochastic matrices in the

3-dimensional affine space{(a$\mathrm{a}_{11}$, a 12. a 13),

$11+\mathrm{a}_{12}+\mathrm{a}_{13^{-}}-$

a1$(\mathrm{a}_{12}$of all 3$\mathrm{H}$, a 22 , a 23) ,$(\mathrm{a}_{13}$$\mathrm{x}3$symmetric real maCrices, a 23, a 33)}

$2+\mathrm{a}_{22}+\mathrm{a}23=\mathrm{a}_{13}+\mathrm{a}_{23}+\mathrm{a}_{33^{-}}-1$

such$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$.

We use a non-orthogonal coordinates system$\langle , \rangle$

in$\mathrm{H}$for which:$\{$(1/3, 1/3 , 1/3) , (1/3, 1/3, 1/3), (1/3, 1/3 , $1/3)\}_{--}<0,0,0>,$$\{(0,1 , 0) , (1 , 0,0) , (0,0 , 1)\}$ $–<1$ , $0,0>$ ,$\{(0,0,1) , (0 , 1 , 0) , (1 , 0,0)\}=<-1/2,$ $\Gamma 3/2$ , $0>$ ,$\{(1 , 0,0) , (0 , 0,1), (0,1 , 0)\}--<-1/2,$ $-\sqrt{3}/2,0>$ ,$\{(1 , 0 , 0), (0,1 , 0) , (0,0 , 1)\}$ $=<0$ , $0,$

for some$\mathrm{b}>0$.

$\mathrm{b}>$