25.2 find all possible Jordan Normal Forms of A

Nov 2009
717
133
Wahiawa, Hawaii
$\tiny{25.2}$
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
 
Jun 2013
1,110
590
Lebanon
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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Nov 2009
717
133
Wahiawa, Hawaii
$$\left[\begin{array{rr}
1&0\\0&1\end{array}\right]\quad
\left[\begin{array{rr}1&1\\0&1\end{array}\right]$$
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 ?
 
Last edited:
Jun 2013
1,110
590
Lebanon
\(\displaystyle \left[\begin {array} {cc} 1 & 1 \\ 0 & 1\end {array} \right]\)

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

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

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

\(\displaystyle \left[\begin {array} {cccc}
4 & 1 & 0 & 0 \\
0 & 4 & 0 & 0 \\
0 & 0 & 4 & 0\\
0 & 0 & 0 & 4
\end {array} \right]\)
 
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Nov 2009
717
133
Wahiawa, Hawaii
Ok I was having trouble with TEX

So if we put all this together we will have 10 possible canonicial forms?
 
Last edited:
Nov 2009
717
133
Wahiawa, Hawaii
j1.PNG

this is what I finished with to send in as hw

probably not conventional notation but...