46

Possible Duplicate:
Matrix is conjugate to its own transpose

How can I prove that a matrix is similar to its transpose?

My approach is: if $A$ is the matrix then $f$ is the associated application from $K^n\rightarrow K^n$. Define $g:K^n\rightarrow (K^n)^*$ by $g(e_i)=e_i^*$, and define $f^T$ to be the transpose application of $f$. I proved that $f^T=gfg^{-1}$. What I don't understand is, what is the matrix associated to $g$, so I can write $A^T=PAP^{-1}$.

Community
  • 1
Mec
  • 5,257

2 Answers2

59

Consider...

$$B^{-1} = B = \begin{bmatrix} 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 1 & 0 \\ \vdots & \ \vdots & & \vdots & \vdots \\ 0 & 1 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \end{bmatrix} \qquad \mathrm{and} \qquad J = \begin{bmatrix} \lambda & 1 & & \\ & \ddots & \ddots & & \\ & & \lambda & 1 \\ & & & \lambda \end{bmatrix} $$

Then $B^{-1}J^TB = J$. Thus a Jordan block $J$ and its transpose $J^T$ are similar. So using $B_1,\dots B_\ell$ for each Jordan block $J_1,\dots,J_\ell$ and letting $$B = \begin{bmatrix} B_1 & & & \\ & B_2 & & \\ & & \ddots & \\ & & & B_\ell \end{bmatrix} \qquad \mathrm{and} \qquad J = \begin{bmatrix} J_1 & & & \\ & J_2 & & \\ & & \ddots & \\ & & & J_\ell \end{bmatrix}$$ Then $B^{-1}J^TB=J$. Therefore, a Jordan form and its transpose are similar.

Finally, put $A$ into its Jordan form: $P^{-1}AP=J$ then $J^T = (P^{-1}AP)^T=P^TA^T(P^T)^{-1}$ so thus $A$ is similar to $J$. $J$ is similar to $J^T$ and $J^T$ is similar to $A^T$. Hence by transitivity $A$ and $A^T$ are similar.

Edit: What if our field is not algebraically closed? [From the comments.]

Of course, we only have a Jordan form when our characteristic polynomial splits over our field. Thus if we are working over a field such as $\mathbb{R}$ or $\mathbb{Q}$, my quick argument does not apply. Here is a fix:

Say we are working over a field $\mathbb{F}$. First, tack on all of the eigenvalues and call that field $\mathbb{K}$ (i.e., $\mathbb{K}$ is the splitting field for our characteristic polynomial).

By the above answer, using Jordan form, $A$ is similar to $A^T$ (working over $\mathbb{K}$). Therefore, $A$ and $A^T$ share the same rational canonical form. Since a matrix is similar to its rational canonical form over the field generated by its entries, $A$ and $A^T$ are similar to this common form working over $\mathbb{F}$ (or possibly even a smaller field). Therefore, $A$ and $A^T$ are similar over $\mathbb{F}$.

Bill Cook
  • 29,244
  • 21
    Doesn't this require that $K$ is algebraically closed? To get around this technicality, one needs to prove that $A$ and $B$ are similar over $K$ if and only if $A$ and $B$ are similar over $K'$ (some field extension of $K$). But essentially, all we need is that $A$ and $B$ are similar if and only if they have the same rational canonical form, and that relation will not change when we consider it over a field extension. – Nicholas Stull Mar 31 '14 at 19:51
  • 2
    Yes. One possible (unsatisfying) repair to extend to nonclosed fields is to notice that since they are similar in an extension field, their invariant factors match. Thus they share the same rational canonical form and thus are similar over the base field. – Bill Cook Mar 31 '14 at 22:50
  • 1
    Dumb question, but why do we know that $J^T$ is similar to $A^T$? $J^T$ isn't necessarily in Jordan normal form, right? So it isn't necessarily the Jordan normal form of $A^T$? – Chill2Macht May 28 '16 at 04:48
  • 1
    The matrix P gives the similarity between A and J. And as addressed in my answer P's inverse transpose gives the similarity between A and J's transposes. – Bill Cook May 29 '16 at 20:14
  • 1
    As for J transpose, no it's not the Jordan form of A transpose (according to my convention) but that's not important. I'm just using the Jordan form as a way station to get something easy to manipulate. That's essentially why Jordan form is useful/of interest. By the way, whether you consider J or J transpose the Jordan form of A is a matter of taste/convention. When constructing P, depending on the way you arrange your "chains" of generalized eigenvectors you'll get either J or J transpose. So actually both are Jordan forms for A (depending on your convention). – Bill Cook May 29 '16 at 20:32
13

Note that reversing the basis order (conjugating by the matrix with ones from bottom left to top right and zeros elsewhere) takes a Jordan block to its transpose, e.g.

$$ \left( \begin{array}{ccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array} \right) \left( \begin{array}{ccc} a&1&0\\ 0&a&1\\ 0&0&a\\ \end{array} \right) \left( \begin{array}{ccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array} \right)= \left( \begin{array}{ccc} a&0&0\\ 1&a&0\\ 0&1&a\\ \end{array} \right). $$

yoyo
  • 9,943