# 25.2 find all possible Jordan Normal Forms of A

#### bigwave

$\tiny{25.2}$
Suppose that A is a matrix whose characteristic polynomial is $$(\lambda-4)^4(\lambda-1)^2$$
find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).

ok I presume this is going to start with $6\times 6$ matrix

#### Idea

yes, there is a $$\displaystyle 2\times 2$$ matrix of which there are 2 since we have 2 integer partitions of 2

and a $$\displaystyle 4\times 4$$ with 5 different matrices since we have 5 integer partitions of 4

for a total of 10

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#### bigwave

$$\left[\begin{array{rr} 1&0\\0&1\end{array}\right]\quad \left[\begin{array{rr}1&1\\0&1\end{array}\right]$$
And
$$\left[\begin{array}{rr} 4&0&0&0\\ 0&4&0&0\\ 4&0&4&0\\ 0&0&0&4\\ \end{array}\right]\quad \left[\begin{array}{rr} 4&1&0&0\\ 0&4&0&0\\ 4&0&4&0\\ 0&0&0&4\\ \end{array}\right]$$
And ?

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#### Idea

$$\displaystyle \left[\begin {array} {cc} 1 & 1 \\ 0 & 1\end {array} \right]$$

$$\displaystyle \left[\begin {array} {cccc} 4 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 4 \end {array} \right]$$

$$\displaystyle \left[\begin {array} {cccc} 4 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end {array} \right]$$

$$\displaystyle \left[\begin {array} {cccc} 4 & 1 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 4 \end {array} \right]$$

$$\displaystyle \left[\begin {array} {cccc} 4 & 1 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0\\ 0 & 0 & 0 & 4 \end {array} \right]$$

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#### bigwave

Ok I was having trouble with TEX

So if we put all this together we will have 10 possible canonicial forms?

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#### bigwave

this is what I finished with to send in as hw

probably not conventional notation but...