Talk:Sylvester's matrix theorem

From Wikipedia, the free encyclopedia

[edit] Normalization

I think the article should explain how to normalize the eigenvectors. -- Jitse Niesen (talk) 14:27, 5 January 2007 (UTC)

In context, the normalization is "obviously" such that r_ic_j = δ_ij, (where δ is the Kronecker delta), but a reference is still required. I think I have one, but it's not called "Sylvester's" in that reference. If the eigenvalues are distinct, then it's easy to construct such normalized eigenvectors, as if . — Arthur Rubin | (talk) 15:00, 5 January 2007 (UTC)

The reference I thought was there didn't have it. However, the actual results seems to be the following:

Part 2: If

and

r_ic_j = δ_ij (where δ is the Kronecker delta)

then

Part 1: (How to get to the hypothesis)

(Option A) If A is diagonalizable, A = UDU ^{− 1}, then, taking:

r_i to be the rows of U,

c_i to be the columns of U^-1,

λ_i to be the diagonal entries of D

the hypothesis of Part 2 can easily be seen to be met.

(Option B, which may be where Sylvester got into it). If A has n distinct (left-)eigenvalues, denote them λ_i.

Let r_i be the corresponding row-eigenvectors

Let d_i be the corresponding column-eigenvectors.

Let

(It can easily be seen that r_i c_j = 0 for i <> j.)

I'm still not convinced that Sylvester did it. — Arthur Rubin | (talk) 21:43, 5 January 2007 (UTC)

[edit] Added "disputed" tag

It seems only to apply (as written) if the matrix is diagonalizable with distinct eigenvalues, and can be adapted only if the matrix is diagonalizable. I'll get back to it, later, but there may very well be a problem. — Arthur Rubin | (talk) 14:37, 5 January 2007 (UTC)