       Re: help in constructing a binomial consequence

• To: mathgroup at smc.vnet.net
• Subject: [mg98265] Re: help in constructing a binomial consequence
• From: mark mcclure <mcmcclur at unca.edu>
• Date: Sun, 5 Apr 2009 06:31:31 -0400 (EDT)
• References: <gr4jm9\$ac5\$1@smc.vnet.net>

```On Apr 3, 5:08 am, Galina <Galina.Pil... at gmail.com> wrote:
> I need  to find an eigenvalues of the matrix M [N*N] where elements
> are of the type F[i]*F[j]. I need help to construct the elements F[i]
> which must be in the following order: F=1, F=x, F=y, F
> =x^2, F=x*y, F=y^2, F=y*x^2, F=x*y^2, F=x^3 an=
d etc....

Your pattern seems a bit off.  I'm guessing you mean:
1, x, y, x^2, x y, y^2, x^3, x^2 y, x y^2, y^3

In this case, you can define your F like so:

biRow[n_] := Table[x^(n - k)*y^k, {k, 0, n}];
Clear[F];
m = 0;
Do[F[++m] = binom,
{binom, Flatten[Table[biRow[n], {n, 0, 3}]]}];

Then, you can define your matrix via:
M = Table[F[i]*F[j], {i, 1, m}, {j, 1, m}];

I'm guessing these matrices all have rank 1.
Thus, you can find the only non-zero eigenvalue
via:
Eigenvalues[M] // Last

Mark McClure

```

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