数理解析研究所講究録
第 868 巻 1994 年 52-65

52Some Recent Results on Isospectral FlowsGEN HORI(堀玄)

Department of Mechano Informatics
The University of Tokyo, Tokyo 113, Japan

Abstract
The following two results related to recent studies on isospectral flows
are presented. (i) The multiple bracket generalizations of Brockett’s double bracket equation $\dot{H}=[H, [H, N]]$ are proved to have similar properties
to Brockett’s equation. (ii) The general formula of isospectral gradient flows,
through which we can obtain useful examples, is derived. A new dynamicalsystem which provides a new computational algorithm for solving the eigenvalue problem of non-symmetric matrices is calculated as an example.

1

Introduction

Matrix dynamical systems whose solutions evolve with their eigenvalues preserved arecalled “isospectral flows”. Recently, isospectral flows have been extensively studied
because they are related to the theory of solitons and the eigenvalue problem of
matrices. The Toda equation[3][4] which is fundamental in the theory of solitons canbe represented as an isospectral flow on the set of tridiagonal matrices [5]. Further,it has been proved that the Toda flow converges to a diagonal matrix whose diagonal
entries are sorted with regard to their absolute values. P.Deift et al. suggested theuse of the Toda flow as a computational algorithm for solving the eigenvalue problem
of symmetric matrices[6].
Brockett[l] introduced the dynamical system on the set of real $n\cross n$ matrices
which is expressed in the double bracket form$\dot{H}(t)=[H(t), [H(t), N]]$(1)

where $[A, B]=AB-BA$ , $N$ is a fixed real diagonal matrix with distinct diagonal
entries, and $H(O)$ is a real symmetric matrix. He proved that (i) the dynamical

53
system is an isospectral flow on the set of real symmetric matrices and (ii) the solutionof the system converges to a diagonal matrix and the diagonal entries of $H(\infty)$ and$N$ are similarly ordered. He also showed that we can sort lists, diagonalize matrices,and solve linear programming problems using this dynamical system. It has beenshown that Brockett’s dynamical system includes the Toda equation and the Riccati
equation as special cases[7].
There are two main purposes of this paper. One is to generalize Brockett’s double
bracket equation to a multiple bracket equation$\dot{H}=\sim^{[H,N]\cdots]]}[H,$$[Hm- fold’$and show that this generalized equation has properties similar to Brockett’s equation
when the multiplicity $m$ is an even number. The other is to derive the general formulaof isospectral gradient flows and obtain useful examples of isospectral flows using theformula.
This paper is organized as follows. In section 2, we generalize Brockett’s double
bracket equation to the multiple bracket equations. In section 3, we derive the generalformula of isospectral gradient flows and present some useful examples obtained
through the use of this formula. The final section contains concluding remarks.

2

Multiple Bracket Equations

2.1

Simple generalization

In this section, we investigate the dynamical systems of the multiple bracket form$N$$\dot{H}(t)=\frac{[H(t),[H(t),\cdots,[H(t)}{m- fo1d},$]$\cdots$]],(2)

matrix and $N$ is a fixed realmatrix.
where $H(t)$ is a realMultiple bracket equations are classified into two families according to the convergence properties of their solutions, that is, into the families of even and odd
multiplicities. The equations whose multiplicity $m$ is even have very similar properties to Brockett’s double bracket equation, but the equations whose multiplicity$m$ is odd have completely difTerent convergence properties from Brockett’s equation.
The properties of the solutions of (2) in the case that $N$ is a diagonal matrix with
distinct diagonal entries and $H(O)$ is symmetric are summarized as follows.
$n\cross n$$n\cross n$

54
is even, (2) defines an isospectral flow on the set of real symmetric matrices
and the solution of (2) converges to the diagonal matrix whose diagonal entries are
similarly ordered to the diagonal entries of $N$ .
If is odd, (2) defines an isospectral flow but has no asymptotically stable fixed
point.
The rest of this section constructs the proof for the above mentioned properties.
1 If $m$$\bullet$$m$

Suppose that $N$ and $H(O)$ are symmetric and $m$ is even. Then
Theorem 2.1
the ordinary differential equation (2) defines an isospectral flow on the set of real
symmetric matrices. The solution $H(t)$ of (2) exists for all $t\in R$ .
Proof. Let $Sym(n, R)$ and Skew $(n, R)$ denote the set of all realand skew-symmetric matrices respectively,$Sym(n, R)$$=$$\{X\in M(n, R)|X^{t}=X\}$ ,Skew $(n, R)$$=$$\{X\in M(n, R)|X^{t}=-X\}$ .

$n\cross n$symmetric

By using the following two obvious facts$X\in Sym(n, R),$ $Y\in Sym(n, R)$$\Rightarrow$[X, $Y$ ] $\in Skew(n, R)$ ,$X\in Sym(n, R)$ , $Y\in Skew(n,R)$$\Rightarrow$

[X,

$Y$

] $\in Sym(n, R)$ ,we get$\underline{[H(t),[H(t),\cdots,[H(t)},$$N$]$\cdots$

]]

$\in$

$Sym(n, R)$( $m$ : even),

(3)

$N$

]

$\cdots$

]]

$\in$

Skew $(n, R)$( $m$ : odd),(4)m-fold$\frac{[H(t),[H(t),\cdots,[H(t)}{m- fo1d},$where $H(t)$ and $N$ are symmetric. From (3), it is immediately obtained that theis even.
solution $H(t)$ of (2) evolves on the set of real symmetric matrices where
Let thematrixbe the solution of the ordinary differential equation$m$$n\cross n$$\Theta(t)$$0(t)=\Theta(t)A(t),$$\Theta(0)=I_{n}$

,

where
$A(t)=\underline{[H(t),[H(t),\cdots,[H(t)},$$N$$(m-1)$ -foldWe can readily verify that
$H(t)=\Theta(t)^{-1}H(0)\Theta(t)$]$\cdots$

]].

(5)

55
satisfies (2) and hence $H(t)$ is isospectral to $H(O)$ for all $t\in R$ . Further, from (4),is even and thus
we see that $A(t)$ defined by (5) is skew-symmetric when
exists for all $t\in R$ and thus $H(t)$evolves on $SO(n)$ . Since $SO(n)$ is compact,
exists for all $t\in R$ .
$m$$\Theta(t)$$\Theta(t)$$\square$Suppose that $N$ is a diagonal matrix with distinct diagonal entries
Lemma 2.1
and $H$ is diagonalizable. Then the following two statements are equivalent for $m=$
$1,2,3,$$\cdots$

(a)
(b)

.

$\underline{[H,[H,\cdots,[H},$$N$]$\cdots$]] $=0$ .

m-fold$H$

is a diagonal matrix.

is obvious. We shall prove.
Suppose that $H$ is a diagonalizable matrix with eigenvalues

Proof.

$(b)\Rightarrow(a)$multiplicities$\lambda_{1}=.$$n_{1},$..

$(a)\Rightarrow(b)$$\cdots,$ $n_{r}( \sum_{i=1}^{r}n_{i}=n)$$=\lambda_{n_{1}}$,$\lambda_{n_{1}+1}=.$.

$,$$\lambda_{1},$$\cdots$,$\lambda_{n}$occurring with, that is,$=\lambda_{n_{1}+n_{2}}$,$\cdot$.. ,$\lambda_{n_{1}+\cdots+n_{r-1}+1}=.$..

$=^{r}\lambda_{n}$.

Further, let us assume that $P\in GL(n, C)$ diagonalizes $H$ so that,$P^{-1}HP$$=$$diag(\lambda_{1}, \cdots, \lambda_{n})$

$=$

diag

.
$( \frac{\lambda_{n_{1}},\cdots,\lambda_{n_{1}}}{n_{1}},\frac{\lambda_{n_{1}+n_{2}},\cdots,\lambda_{n_{1}+n_{2}}}{n_{2}}, \cdots,\sim_{n_{r}}^{n}\lambda_{n}, \cdots, \lambda)$

By multiplying both sides of (a) by$P^{-1}$on the left and by $P$ on the right, we get$P^{-1}NP$$\frac{[P^{-1}HP,[P^{-1}HP,\cdots,[P^{-1}HP}{m- fo1d},$]$\cdots$]] $=0$ .

is a real] $=(a_{i}-a_{j})E_{ij}$ where
Observing the fact that [diagmatrix whose $(i, j)$ -th entry is 1 and all the other entries are , we see thatwhere
is
the $(i, j)$ -th entry of the left hand side of the equation is
the $(i, j)$ -th entry of $P^{-1}NP$ . It follows that $P^{-1}NP$ is a block diagonal matrix,$(a_{1},$ $a_{2},$$a_{n}),$ $E_{ij}$$\cdots,$$E_{ij}$$0$$n\cross n$$(\lambda_{i}-\lambda_{j})^{m}n_{ij}’$$P^{-1}NP=diag(B_{1}, \cdots, B_{r}),$ $B_{i}\in M(n_{i}, C)$$n_{ij}’$.

is a direspectively, namely
Assume that $P_{i}\in GL(n_{i}, C)$ diagonalizes. Suchareexist because the eigenvalues ofagonal matrix for $i=1,2,$$B_{i}$$\cdots,$ $r$$P_{i}’ s$$P_{i}B_{i}P_{i}^{-1}$$B_{i}’ s$56
distinct since diagonal entries of $N$ are assumed to be distinct. We can choose apermutation matrix so that$Q$.

$Qdiag(P_{1}, \cdots, P_{r})P^{-1}NPdiag(P_{1^{-1}}, \cdots, P_{r}^{-1})Q^{-1}=N$

This equation is equivalent to$[N, Pdiag(P_{1}^{-1}, \cdots, P_{r}^{-1})Q^{-1}]=0$.

From this and the assumption that $N$ is a diagonal matrix with distinct diagonalmust be a diagonal matrix,
entries, Pdiag$(P_{1^{-1}}, \cdots, P_{r^{-1}})Q^{-1}$( $D$ : diagonal matrix).

$D=Pdiag(P_{1^{-1}}, \cdots, P_{r}^{-1})Q^{-1}$Now we conclude that$P$

must have the form

$P=DQdiag(P_{1}, \cdots, P_{r}),$ $P_{i}\in GL(n_{i}, C)$

where $Q\in GL(n, C)$ is a permutation matrix and $D\in GL(n, C)$ is a diagonal matrix.
Then $H$ can be expressed as$H=P(P^{-1}HP)P^{-1}$$=DQ^{\sim}diag(P_{1}, \cdots, P_{r})diag(\lambda_{n_{1}}, \cdots, \lambda_{n}\lambda_{n}, \cdots, \lambda_{n})diag(P_{1^{-1}}, \cdots, P_{r^{-1}}\vee^{1}’\cdots,\sim)Q^{-1}D^{-1}$$n_{r}$$n_{1}$$=DQdiag(\lambda_{n_{1}}, \cdots, \lambda\lambda_{n}, \cdots, \lambda)Q^{-1}D^{-1}\sim_{n_{1}}^{n_{1}}’\cdots,\sim_{n_{r}}^{n}$

It follows that $H$ is a diagonal matrix.

$\square$Suppose that $N$ is a diagonal matrix with distinct diagonal entries,
Theorem 2.2$H(O)$ is symmetric and $m$ is even.
Then for the solution $H(t)$ of (2), $H(\pm\infty)=$
exists and is a diagonal matrix.

$\lim_{tarrow\pm\infty}H(t)$Observing the fact that $tr(A[B, C])=tr([A, B]C)$ for any real $n\cross n$
Proof.
matrices $A,$ $B,$ , we have$C$$\frac{d}{dt}tr(NH)=tr(N\underline{[H,[H,\cdots,[H}, N]\cdots]])$m-fold$=$$tr([N, H]\underline{[H,[H,\cdots,[H}, N]\cdots]])=(-1)tr([H, N]\underline{[H,[H,\cdots,[H}, N]\cdots]])$$(m-1)$ -fold$(m-1)$ -fold$=$$(-1)tr([[H, N], H]\underline{[H,[H,\cdots,[H}, N]\cdots]])=(-1)^{2}tr([H, [H, N]]\underline{[H,[H,\cdots,[H}, N]\cdots]])$$(m-2)$ -fold$=$$(m-2)$ -fold]$=(-1)^{m/2}tr(\underline{[H,[H,\cdots,[H}, N$

$m/2$

-fold

$\cdots$]]$\underline{[H,[H,\cdots,[H},$

$m/2$

-fold

$N$

]

$\cdots$]]).

57
Further, since $[A, B]^{t}=-[A, B^{t}]$ where$\underline{[H,[H,\cdots,[H},$

$m/2$

$N$

]

$\cdots$]$]^{t}$$=$$A$

is symmetric, we get$(-1)[H, [H, \cdots, [HN]\cdots]^{t}]\sim$
’

-fold

$m/2$
$=$

-fold

$=(-1)^{m/2}\underline{[H,[H,\cdots,[H},$ $N$

$m/2$

]

$\cdots$

]],

-fold

which yields$\frac{d}{dt}tr(NH)=tr(\underline{[H,[H,\cdots,[H}, N]\cdots]]\underline{[H,[H,\cdots,[H},$ $N$

$m/2$

-fold

$m/2$

]

$\cdots$])$]^{t}$$\geq 0$.

-fold

Here we used the fact that $tr(AA^{t})=\sum_{i,j}a_{ij}^{2}\geq 0$ for any real $n\cross n$ matrix

$A$

and

the equality holds if and only if $A=0$ . Thus $tr(NH)$ is monotone increasing, and isbounded from both below and above because it is a continuous function on a compact
set. Therefore it converges as $tarrow\pm\infty$ and its derivatives goes to zero. From theabove calculation, $\frac{d}{dt}tr(NH)=0$ holds if and only if$\underline{[H,[H,\cdots,[H},$$N$]$\cdots$]] $=0$ and-fold
is diagonal.
$m/2$

from Lemma 2.1, this is satisfied if and only if$H$

Suppose that $N$ is a diagonal matrix diag
Theorem 2.3
diagonal entries, $H(O)$ is a symmetric matrix with eigenvalues$\square$$(\mu_{1}, \cdots , \mu_{n})$with multiplicities$n_{1},$$\cdots,$$\lambda_{1}=\cdots=\lambda_{n_{1}},$$n_{r}( \sum_{i=1}^{r}n_{i}=n)$$\cdots,$$\lambda_{n_{1+1}}=\cdots=\lambda_{n_{1}+n_{2}}$, .. ,$\cdot$$\lambda_{n_{1}+\cdots+n_{\Gamma}-1+1}=\cdots=\lambda_{n}$$\frac{n!}{n_{1}!\cdots n_{r}!}$where is some permutation on
diagof the linearization of (2) at each fixed point are$\pi$$-(\lambda_{\pi(i)}-\lambda_{\pi(j)})^{m-1}(\mu_{i}-\mu 4)$$n$$\cdots,$,

fixed points of the form

letters and the eigenvalues$(i,j=1, \cdots, n)$.

Thus exactly one of these fixed points is asymptotically stable, whereare similarly ordered.

$\lambda_{\pi(1)},$$\lambda_{n}$, that is,

and $m$ is even. Then the dynamical system (2) has$(\lambda_{\pi(1)}, \cdots , \lambda_{\pi(n)})$$\lambda_{1},$with distinct
occurring$\mu_{1},$$\cdots,$$\mu_{n}$and$\lambda_{\pi(n)}$Proof.
From Lemma 2.1, all the fixed points of the dynamical system (2) arediagonal matrices and their diagonal entries are some permutation of the eigenvalues

58
of $H(O)$ because of the isospectral property.
By using the relations
operator , and [diag$\partial$$\frac{\partial}{\partial h_{ij}}H=E_{ij},$$(a_{1},$ $a_{2},$$\cdots,$

for any differential$\partial[A, B]=[\partial A, B]+[A, \partial B]$$a_{n}),$ $E_{ij}$] $=(a_{i}-a_{j})E_{ij}$ , we obtain$\frac{\partial}{\partial h_{ij}}N]\cdots]]\frac{[H,[H,\cdots,[H}{m- fo1d},|_{H=diag(\lambda_{\pi(1)},\ldots,\lambda_{\pi(n)})}$

$=$

$N$
$\{_{\frac{[E_{ij},[H,\cdots,[H}{m- fo1d}},$]$\cdots$

]]

$+N$
,

]

$\cdots$]]$\frac{[H,[E_{ij},\cdots,[H}{m- fo1d}$$N$]$\cdots$]]$\}|_{H=diag(\lambda_{\pi(1)},\cdots,\lambda_{\pi(n)})}$$+ \cdots+\frac{[H,[H,\cdots,[E_{ij}}{m- fold},$$N$]$\cdots$]]$|_{H=diag(\lambda_{\pi(1)},\ldots,\lambda_{\pi(n)})}$$= \frac{[H,[H,\cdots,[E_{ij}}{m- fold},$$=$$E_{ij}$]$\cdots$]]$|_{H=diag(\lambda_{\pi(1)},\cdots,\lambda_{\pi(n)})}$$- \frac{[H,[H,\cdots,[N}{m- fo1d},$$=$$-(\lambda_{\pi(i)}-\lambda_{\pi(j)})^{m-1}(\mu_{i}-\mu j)E_{ij}$.

This equation can be read as a characteristic equation with an $eigenvalue-(\lambda_{\pi(i)}-$, showing that all the eigenvalues of theand an eigenvector
have non-positive
linearization of (2) at the fixed point
real parts if and only if,
andare similarly ordered. Theorem
2.2 also guarantees that the dynamical system (2) has at least one asymptoticallystable fixed point if $m$ is even. Thus exactly one of the fixed points is asymptoticallyare similarly ordered.
stable, where,and$\lambda_{\pi(j)})^{m-1}(\mu_{i}-\mu_{j})$$E_{ij}$$H=diag(\lambda_{\pi(1)}, \cdots, \lambda_{\pi\langle n)})$$\mu_{1},$$\mu_{1},$$\cdots$$\mu_{n}$$\cdots$$\mu_{n}$$\lambda_{\pi(1)},$$\lambda_{\pi(1)},$$\cdots,$$\lambda_{\pi(n)}$$\cdots,$$\lambda_{\pi\langle n)}$$\square$We can also obtain the following corollary by giving $m$ an odd value in thecalculations used in the proofs of Theorems 2.1 and 2.3.
Corollary 2.1 If $m$ is an odd number,(a) the dynamical system (2) defines an isospectral flow,(b) the solution $H(t)$ of (2) leaves $Sym(n, R)$ even if $H(O)$ is symmetric, and(c) the dynamical system (2) has no asymptotically stable fixed point if $H(O)$ is diagonalizable.

59
2.2

Extended generalization

In this section, we present an example of a little more generalized bracket equations
which have different convergence properties from the equations introduced in theabove section,$\dot{H}(t)=[H(t), [H(t), [N, [H(t), N]]]]$(6)
where $N$ is a diagonal matrix with distinct diagonal entries and $H(O)$ is a symmetricmatrix. That (6) defines an isospectral flow on the set of symmetric matrices isproved by the same way as the proof of Theorem 2.1. By the same calculation as
used in the proof of Theorem 2.3, the eigenvalues of the linearization of (6) at thearediagonal point$H=diag(\lambda_{\pi(1)}, \cdots, \lambda_{\pi(n)})$$-(\lambda_{\pi(i)}-\lambda_{\pi(j)})^{2}(\mu_{i}-\mu_{j})^{2}$$(i,j=1, \cdots, n)$ ,and it follows that all the diagonal points are exponentially stable fixed points if
the initial matrix $H(O)$ has distinct eigenvalues. On the other hand, it is uncertain
whether or not (6) has fixed points other than diagonal points and whether the global
convergence of (6) to diagonal points is guaranteed.
We can easily construct many examples of multiple bracket equations with various
convergence properties by the same way.

3

Isospectral Gradient Flows

3.1

Derivation of the equation

Let $G\subset GL(n, C)$ be a linear Lie group and 9 its Lie algebra. The adjoint orbit ofwhich passes throughis the subset of 9 $l(n, C)$ defined as$G$$A_{0}\in \mathfrak{g}\mathfrak{l}(n, C)$$\Omega_{A_{0}}^{G}=\{g^{-1}A_{0}g|g\in G\}$.

Now we can say that “isospectral flows” are the flows on adjoint orbits.
In this section, we derive the equation of the gradient flow on the adjoint orbitof the real linear Lie group $G\subset GL(n, R)$ which passes through the real matrixof an arbitrary potential function : $gI(n, R)arrow R$ .
Let the inner product of the Lie algebra $gl(n, R)$ be defined as$\Omega_{A}^{G_{0}}$$A_{0}\in \mathfrak{g}\mathfrak{l}(n, R)$$\psi$$\mu(X, Y)=tr(X^{t}Y)$ .

60
Then, for arbitrary $X,$

$Y,$ $Z\in g\mathfrak{l}(n, R)$,(7)$\mu(X, [Y, Z])=-\mu([X, Y^{t}], Z)$

holds.
An arbitrary element in the neighborhood of the identity element ofexpressed as $g=e^{X}$ with $X\in 9$ . Observing that
$g$

$G$

can be$e^{-tX}Ae^{tX}=A+t[A, X]+O(t^{2})$and $X$ , we see that an arbitrary element of the tangent
for arbitrary matrices
atspace of the adjoint orbit
can be expressed as $[A, X]$ with $X\in 9$ .
Let
be the -orthonormal basis of 9. The directed derivative of $\psi(A)$ at
along the direction $[A, X_{k}]$ is$A$$\Omega_{A_{0}}^{G}$$\{X_{k}\}$$A\in\Omega_{A_{0}}^{G}$

$A$

$\mu$

.

$\sum_{i,j}\frac{\partial\psi}{\partial a_{ij}}[A, X_{k}]_{ij}=\mu((\frac{\partial\psi}{\partial a_{ij}})[A, X_{k}])$The steepest descent direction of $\psi(A)$ which is the negative of the linear combinationof $[A, X_{k}]s$ weighted by these coefficients is calculated as follows using (7):$- \sum_{k}\mu((\frac{\partial\psi}{\partial a_{ij}}), [A, X_{k}])[A, X_{k}]$$=$$-[A, \sum_{k}\mu((\frac{\partial\psi}{\partial a_{ij}})[A, X_{k}])X_{k}]$$=$$[A, \sum_{k}\mu([(\frac{\partial\psi}{\partial a_{ij}}), A^{t}],X_{k})X_{k}]$$=$$[A, \pi_{9}[(\frac{\partial\psi}{\partial a_{ij}})A^{t}]]=-[A, \pi_{9}[A^{t}, (\frac{\partial\psi}{\partial a_{ij}})]]$

,

where
is the -orthogonal projection from $9^{[}(n, R)$ to its subspace 9. Then thegradient equation is$\pi_{9}$$\mu$$\frac{dA(t)}{dt}=-[A(t), \pi_{9}[A(t)^{t}, (\frac{\partial\psi}{\partial a_{ij}}I ]]$

3.2

.

(8)

The convergence of the flow

In the case that the potential function $\psi(A)$ is bounded below,
exists for the solution $A(t)$ of (8).

$A( \infty)=\lim_{tarrow\infty}A(t)$61
The time-derivative of $\psi(A(t))$ along the orbit $A(t)$ is calculated as follows using(7):$\frac{d}{dt}\psi(A(t))=\sum_{i,j}\frac{\partial\psi}{\partial a_{ij}}\frac{da_{ij}}{dt}$Because$\dot{\psi}arrow 0(tarrow\infty)$$=$$- \sum_{i,j}\frac{\partial\psi}{\partial a_{ij}}[A, \pi_{9}[A^{t}, (\frac{\partial\psi}{\partial a_{ij}})]]_{ij}$$=$$- \mu((\frac{\partial\psi}{\partial a_{ij}})[A, \pi_{9}[A^{t}, (\frac{\partial\psi}{\partial a_{ij}})]])$$=$$\mu([(\frac{\partial\psi}{\partial a_{ij}}), A^{t}], \pi_{9}[A^{t}, (\frac{\partial\psi}{\partial a_{tj}})])$$=$$- \mu([A^{t}, (\frac{\partial\psi}{\partial a_{ij}})], \pi_{9}[A^{t}, (\frac{\partial\psi}{\partial a_{ij}})])$.

, we conclude that(9)$\pi_{9}[A(\infty)^{t}, (\frac{\partial\psi}{\partial a_{ij}})]=0$

is satisfied at $A(\infty)$ .

3.3

Flows on the set of complex matrices

In this section, the discussions in the above sections are generalized for the adjoint
orbit
of $G\subset GL(n, C)$ which passes through $A_{0}\in 9^{[}(n, C)$ and a potentialfunction: $gl(n, C)arrow R$ . Note that we can’t use complex differentiation for theis a:potential function
since it is singular (excepting whenconstant function). In the derivation of the gradient equation, we treat $9^{[}(n, C)$ notas the -dimensional linear space on , but as the-dimensional linear space on$\Omega_{A_{0}}^{G}$$\psi$$\psi$$\mathfrak{g}\mathfrak{l}(n, C)arrow R$$2n^{2}$$C$$n^{2}$$R$$\psi$.

We state the results without proof. Letandbe the real and imaginary partof the $(i, j)$ -th entry of respectively. We define the inner product of $9^{[}(n, C)$ as$a_{ij}$$b_{ij}$$A$$\mu(X+\tilde{X}i, Y+\tilde{Y}i)=tr(X^{t}Y+\tilde{X}^{t}\tilde{Y})$where$X,\tilde{X},$ $Y,\tilde{Y}$,are real matrices. Then the gradient equation is,$\frac{dA(t)}{dt}=-[A(t), \pi_{9}[A(t)^{*}, (\frac{\partial\psi}{\partial a_{ij}}+\frac{\partial\psi}{\partial b_{ij}}i)]]$

(10)

and
$\pi_{9}[A(\infty)^{*}, (\frac{\partial\psi}{\partial a_{ij}}+\frac{\partial\psi}{\partial b_{ij}}i).]=0$

is satisfied at$A(\infty)$

.

(11)

62
3.4

Previous examples

In this section, we derive two already known isospectral flows as special cases of (8)and (10).
1. Brockett[l]’s flow.
Let $G=SO(n),$ $\psi(A)=-tr(AN)$ where $N$ is a real constant symmetric matrix and$A(O)$ be a real symmetric matrix in (8). Then we get the Brockett’s equation (1)$A(t)=[A(t), [A(t), N]]$ .

(12)Note that we can omitin this equation because $[A(t), N]\in 5o(n)$ where $A(t)$and $N$ are symmetric.
2. The dynamical system that diagonalizes skew-Hermitian matrices.
$\pi_{50\langle n)}$

Let $G=SU(n),$$\psi(A)=\frac{1}{2}\sum_{i\neq j}a_{ij^{2}}+b_{ij}^{2}$

dynamical system on

and $A(O)\in\epsilon u(n)$ in (10). Then we get the$\epsilon u(n)$(13)$\dot{A}(t)=-[A(t), [A(t), J(A(t))]]$ ,where $J(A)$ is the diagonal matrix obtained by replacing all the non-diagonal entries
of by zeros. Note that we can omitin this equation because $[A(t), J(A(t))]\in$
where $A(t)\in 5U(n)$ . It was shown that the solution $A(t)$ of this dynamicalsystem converges to a diagonal matrix for almost all initial values $A(O)\in\epsilon u(n)$ in
Nakamura[8].
$A$$\pi_{5U(n)}$$\epsilon u(n)$3.5

A new dynamical system for the eigenvalue problem of

non-symmetric matrices
In this section, we derive a new dynamical system that non-symmetric complex
matrices as a special case of (10).
Let $G=GL(n, C), \psi(A)=\frac{1}{2}\sum a_{ij^{2}}+b_{ij}^{2}$ and$A(0)\in \mathfrak{g}\mathfrak{l}(n, C)$in (10). Then we$i>j$get the dynamical system on $g\iota(n, C)$$A(t)=-[A(t), [A(t)^{*}, L(A(t))]]$ ,(14)

where $L(A)$ is the matrix obtained by replacing all the upper triangular entries ofby zeros.
It is proved in the following Lemma 3.1 that the fixed point condition (11) for thisdynamical system, namely, $[A(t)^{*}, L(A(t))]=0$ is satisfied if and only if) is an$A$$A\{t$

63
upper triangular matrix. Then the solution $A(t)$ of the dynamical system converges
to an upper triangular matrix for an arbitrary initial value $A(O)$ . This dynamicalsystem provides a new computational algorithm for solving the eigenvalue problem
of non-symmetric complex matrices.
For any complex matrix , the following two statements are equivaLemma 3.1
lent.
(a) $[A^{*}, L(A)]=0$ .
is an upper triangular matrix.
(b)
$A$

$A$

is obvious. To prove
the following two statements.
(i) $a_{i1}=0$ $(2\leq i\leq n)$ .
, $n-1$ ,(ii) For $k=2,3,$

Proof.

$(b)\Rightarrow(a)$$(a)\Rightarrow(b)$by induction, it is enough to show$\cdots$$a_{i1}$$=$$0$$(2\leq i\leq n)$,$a_{i2}$$=$$0$$(3\leq i\leq n)$,$0$$(k\leq i\leq n)$$0$$(k+1\leq i\leq n)$ .

:$a_{i,k-1}$$=$$\Downarrow$$a_{ik}$$=$

The $(1, 1)$ -th entry of $[A^{*}, L(A)]$ is calculated as follows,$[A^{*}, L(A)]_{11}= \sum_{2\leq i\leq n}\overline{a_{i1}}a_{i1}$.

This is enough to show (i). Under the assumptions of ii), the $(k, k)$ -th entry of$[A^{*}, L(A)]$ is calculated as follows,$[A^{*}, L(A)]_{kk}= \sum_{k+I\leq i\leq n}\overline{a_{ik}}a_{ik}$This is enough to show (ii).

4$\square$Concluding Remarks

Brockett[2] already derived results very similar to our results stated in section 3 ofthis paper using the Killing metric instead of the metric . His results is describedin the following theorem.
$\kappa$

$\mu$

64
Theorem 4.1
Let
be a real compact semi-simple Lie group and let 9 be its
Lie algebra. Let $g_{0}\in 9$ and let
be an adjoint orbit. If : $\theta(g_{0})arrow R$ is a
is given bydifferentiable function then the corresponding gradient flow on$G$$\psi$$\theta(g_{0})$$\theta(g_{0})$$\dot{x}=-[x, [x, \psi_{x}]]$.

Along this flow
. As goes to infinity
and approaches an equilibrium point.
$\dot{\psi}=-\langle[x, \psi_{x}],$$[x, \psi_{x}]\}_{\kappa}$

$t$

(15)
$\psi$

approaches a constant$x$

See Brockett[2] for detail. Note that Brockett[l]’s equation (12) and (13) canbe derived from both our formula (10) and Brockett[2]’s formula (15), but our new
dynamical system (14) can not be derived from Brockett[2]’s formula (15) since
is not compact or semi-simple. Our results can be regarded as non-compact
generalization of Brockett[2]’s results.
$\mathfrak{g}\mathfrak{l}(n, C)$

Acknowledgments
The author is very grateful to Professor Shuji Yoshizawa for his valuable advice
and heartfelt encouragement. Also the author is indebted to Professor Yoshimasa
Nakamura for his valuable advice and Mr. Akio Fujiwara for his encouragement andverification of the calculations used in the proofs of the theorems.

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