Difference between revisions of "Jordan Canonical Form"
Tboudreaux (talk | contribs) |
Kfagerstrom (talk | contribs) |
||
| Line 1: | Line 1: | ||
| − | $$ F $$ field, $$V$$ finite dim vector space, $V\xrightarrow{t} V$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $B$ such that $[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$. | + | $$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $$B$$ such that $$[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$$. |
| − | Where each Jordan block $J_n(r) is an $n\times n$ matrix with entries $a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases} | + | Where each Jordan block $$J_n(r)$$ is an $$n\times n$$ matrix with entries $$a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases}$$ |
| − | Each $r_i\in F$ is a root of $c_t$, the characteristic polynomial, and $e_i\geq 1$. The polynomials $(x-r_i)^{e_i}$ are the elementary divisors of $t$, and this jordan canonical form for $t$ is unique up to order of the blocks | + | Each $$r_i\in F$$ is a root of $$c_t$$, the characteristic polynomial, and $$e_i\geq 1$$. The polynomials $$(x-r_i)^{e_i}$$ are the elementary divisors of $$t$$, and this jordan canonical form for $t$ is unique up to order of the blocks |
Revision as of 16:08, 14 April 2023
$$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transforamtion. Assume that the charateristic polynomial of $t$ compleately factor into linear forms (so things that look like $(x-r)^l$). Then there exists a basis $$B$$ such that $$[t]_B^B=\begin{bmatrix} j_{e_1}(x_1)&\\ \; \ddots & \hspace{-12pt} j_{e_n}(x_n) \end{bmatrix}=j(t)$$.
Where each Jordan block $$J_n(r)$$ is an $$n\times n$$ matrix with entries $$a_{ij}=\begin{cases} r, & i=j \\ 1, & j=i+1 \\ 0 & \text{else} \end{cases}$$
Each $$r_i\in F$$ is a root of $$c_t$$, the characteristic polynomial, and $$e_i\geq 1$$. The polynomials $$(x-r_i)^{e_i}$$ are the elementary divisors of $$t$$, and this jordan canonical form for $t$ is unique up to order of the blocks
