Difference between revisions of "Jordan Canonical Form"
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| − | $$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear | + | $$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transformation. Assume that the characteristic polynomial of $$t$$ completely 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 |
Latest revision as of 16:03, 26 April 2023
$$ F $$ field, $$V$$ finite dim vector space, $$V\xrightarrow{t} V$$ linear transformation. Assume that the characteristic polynomial of $$t$$ completely 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
