Isospectral Flow

If A is symmetric, i. e., A є Sn, it has n eigenvalues (X;)n and n corresponding orthonormal eigevectors q; that span Rn. The matrix A may be written

A = QaQt,

where

a = diag(X1,X2,…,Xn), Q = [qt, q2,…, qn], QQT = QT Q = I.

The family of matrices with all possible orthogonal matrices Q forms an isospectral family, the complete family with the given spectrum (Xif[.

Instead of seeking the complete family, we look for a family in which Q depends on a single parameter t.

A(t) = Q(t)AQT(t).

Differentiating w. r.t. t we find

A = qaQ t + Q aqt.

Since Q is orthogonal, we may write

A = (Qaqt)(qq t) + (Q qt)(Qaqt).

Put QQT = S then, since QaQt = A, we have A = AS + ST A.

Now, Q is orthogonal, so that QQT = I, and hence QQT + QQT = 0 : S + ST = 0.

This means that

A = AS – SA

Подпись:and that S is a skew-symmetric matrix. We note that (7) is a so-called autonomous differential equation: the parameter t does not appear explicitly; it appears implicitly because S depends on A, i. e., S = S(A), and A depends on t.

Most importantly, we may argue conversely: if S is skew symmetric and A varies according to (7) then A(t) keeps the same eigenvalues, those for A(0). We may choose S in many ways; different choices will lead to different isospectral families.

The autonomous differential equation (7), called the Toda flow equation, was investigated first for tridiagonal matrices, with the choice

Подпись: a b ■ 0 bi A(t) = b a2 b2 , S(t) = - b1 0 b2 .. .. bn-1 .. .. bn-1 bn — 1 an -bn-1 0 (8)

as in Symes (1982). Watkins (1984) gives a survey of the general theory. See also Chapter 7 of Gladwell (2004). In this case, it may easily be shown that AS – SA is tridiagonal, so that if A(0) is tridiagonal, then A(t) will be tridiagonal. This is a special case of he result that if A(0) is a staircase matrix and if

S(t) = A+(t) – A+T(t), (9)

where A+(t) denotes the upper triangle of A(t), then Equation (7) constrains A(t) to remain a staircase matrix, with the same staircase dimensions as A(0). In general, even if A(0) is a staircase with holes, these holes will eventually be filled in. Gladwell (2002) showed that the Toda flow (7), with S given by (9) maintains the properties TP, NTN, O and SO.

There are two important engineering structures for which the stiffness matrix is a staircase matrix: the rod in logitudinal vibration, already mentioned; an Euler-Bernoulli beam in flexure, for which the stiffness matrix is pentadiagonal, see Gladwell (2002b).

To obtain an isospectral flow for more general cases, we must pose the question: How may we construct an isospectral flow that constrains A to lie on a given graph G?

Consider a very simple graph, the star on n vertices, as shown in Figure 4; a matrix on G has the form A5 in Equation (10); the only non-zero entries are those on the borders, i. e., the first row and column, and the diagonal. The matrix S must be skew-symmetric, so that we need consider only its upper triangle. We choose sij = aij for entries in the first row, and then find the remaining entries below the first row and above the diagonal by making aij = 0 for those entries; there are m = (n – 1)(n – 2)/2 algebraic equations for the m unknown sij.

a1

b2

b3 .

. bn

<N

-ft

О

b3 .

. bn

b1

a2

0

s23 .

. s2n

A5 =

b3

a3

Xfi

U

II

skew

0

. sn-1,n

(10)

_ bn

an

0

The equations for the sij are separable, that for sij is

(ai – aj)sij + 2bibj = 0 i = 2, … ,n, j = i + 1, … ,n. (11)

Подпись: Fig. 4. A star on n vertices.

These m algebraic equations are combined with the equations

n

a = — 2^^ b2, a j = j = 2,…,n (12)

i=2

j — 1 n

bj = (a1 — aj)bj + Sijbi —^2 Sjibi, j = 2,…,n. (13)

i=2 i=j+1

Подпись: bj = bj Подпись: n a1 — aj + 2 k=2 Isospectral Flow Подпись: j = 2,...,n, Подпись: (14)

On substituting for sij from (11) into (13), we find

where’ denotes k = j .It may be shown that if the ai (0), i = 2,… ,n are distinct, and the bi (0), i =

2,. .. ,n are non-zero, then the ai (t),i = 2,… ,n will be distinct, and the bi (t) will be non-zero, so that the denominators in (14) will remain non-zero.

This procedure may be generalised: in the upper triangle of S, take sij = aij when (i, j) є E; find sij for the remaining entries (i, j) Є E by demanding that aij = 0 when (i, j) / E; this gives a set of p algebraic equations for the p entries, sij which are combined with the remaining equations for aij, (i, j) є E, and aii, i = 1, 2,… ,n. The algebraic equations for the sij will have coefficients that are linear combinations of the aij, as in Equation (11) for the star. In general, unlike for the star, we cannot assume that the p equations for the p entries sij will always admit a solution; we will have to fall back on continuity arguments – if they admit a solution when t = 0, they will admit a solution for some small interval of t around t = 0.

The application of this procedure to a typical FEM model, a triangular model of a membrane, is the topic of a forthcoming paper.

References

Gladwell, G. M.L. (1995), On isospectral spring-mass systems, Inverse Problems, 11, 591-602. Gladwell, G. M.L. (1997), Inverse vibration problems for finite element models, Inverse Problems, 13,311-322.

Gladwell, G. M.L. (1998), Total positivity and the QR algorithm, Linear Algebra Appl., 271, 257­272.

Gladwell, G. M.L. (1999), Inverse finite element vibration problems, J. Sound Vibration, 211, 309­342.

Gladwell, G. M.L. (2002a), Total positivity and Toda flow, Linear Algebra Appl., 350, 279-284. Gladwell, G. M.L. (2002b), Isospectral vibrating beams, Proc. Roy. Soc. London A, 458,2691-2703. Gladwell, G. M.L. (2004), Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht. Symes, W. W. (1982), The QR algorithm and scattering for the finite non-periodic Toda lattice, Physica d., 4, 275-280.

Watkins, D. S. (1984), Isospectral flows, SIAM Review, 26, 379-391.