Difference between revisions of "Jordan Canonical Form"
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| − | $$ F $$ 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 18:08, 4 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
