Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix’ (Linear Multilinear Algebra 65 (2017), 1362–1373)

This article refers to:

Every infinite triangular matrix is similar to a generalized infinite Jordan matrix

The author regrets that the printed version of the above article contains some errors in the proof of Proposition 2.3 and would like to apologize for that. Moreover, to fill this gap at least a little bit, the author would like to formulate and briefly sketch the proof of the following.

Lemma 0.7:

Suppose that F is any field and that . Let , , , denote the indices of nonzero columns of a ordered increasingly. If for all the following condition holds

then there exists such that is equal to a generalized .

Proof Additionally, let be equal to .

To prove the following it suffices to go through two steps.

(1)	There exists a matrix such that has the following properties: • if i-th column of a is zero, then i-th column of is zero, • for every i, , and every , , , , we have . The matrix g can be constructed inductively on consecutive nonzero columns.
(2)	The obtained matrix is equal to where has only nonzero coefficients on its first superdiagonal. Such is similar (in ) to .

Obviously, from what is written above it follows that the main result is not necessarily true. However, let us state that from correct results it follows the below statements.

Corollary 0.8:

The set of upper triangular infinite matrices that are similar in to the generalized infinite Jordan matrix is dense in .

Corollary 0.9:

The set of matrices from that are similar in to the generalized infinite Jordan matrix is dense in .