# a question about “triangular matrix”

I read the article from wiki about triangular matrix, it says "A matrix which is conjugate to a triangular matrix is called triangularizable." I do not quite understand: isn't any triangularizable matrix still a triangular matrix, since the conjugate of any triangular matrix is still triangular?

This "conjugate" means "similar to" or something else, or it does not say anything non-trivial here?

Thanks.

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I don't think triangularizable matricies have to be triangular. For instance, diaginal matricies are triangular so any diagonalizable matrix is triangularizable. –  user3180 Apr 22 '11 at 23:11
"Conjugate" means "similar to" in this context. en.wikipedia.org/wiki/Similar_matrix –  Jonas Meyer Apr 22 '11 at 23:13
@Jonas, ah! This "conjugate" terminology means always "transposition"-related to me. It is quite confusing here. :) –  Qiang Li Apr 22 '11 at 23:16
The link is really confusing! –  Qiang Li Apr 22 '11 at 23:23

Here, conjugate is understood in the Algebraic sense: two matrices A and B are conjugates (also said to be similar to each other) if there exists an invertible matrix C s.t. $A = C^{-1} B C$.
I am confused by the sentence "Any square matrix will full rank is triangularizable, for example." Triangularizability depends on the field. E.g., $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ has full rank but is not triangularizable in $M_2(\mathbb R)$. Every matrix, regardless of rank, is triangularizable in $M_n(\mathbb C)$. –  Jonas Meyer Apr 23 '11 at 1:55