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Thread: 25.2 find all possible Jordan Normal Forms of A

  1. #1
    Super Member bigwave's Avatar
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    25.2 find all possible Jordan Normal Forms of A

    $\tiny{25.2}$
    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
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  2. #2
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    Re: 25.2 find all possible Jordan Normal Forms of A

    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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    Super Member bigwave's Avatar
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    Re: 25.2 find all possible Jordan Normal Forms of A

    $$\left[\begin{array{rr}
    1&0\\0&1\end{array}\right]\quad
    \left[\begin{array{rr}1&1\\0&1\end{array}\right]$$
    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 by bigwave; Mar 20th 2019 at 10:45 AM.
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    Re: 25.2 find all possible Jordan Normal Forms of A

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

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

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

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

    $\displaystyle \left[\begin {array} {cccc}
    4 & 1 & 0 & 0 \\
    0 & 4 & 0 & 0 \\
    0 & 0 & 4 & 0\\
    0 & 0 & 0 & 4
    \end {array} \right]$
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  5. #5
    Super Member bigwave's Avatar
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    Re: 25.2 find all possible Jordan Normal Forms of A

    Ok I was having trouble with TEX

    So if we put all this together we will have 10 possible canonicial forms?
    Last edited by bigwave; Mar 20th 2019 at 11:00 AM.
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  6. #6
    Super Member bigwave's Avatar
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    Re: 25.2 find all possible Jordan Normal Forms of A

    25.2 find all possible Jordan Normal Forms of A-j1.png

    this is what I finished with to send in as hw

    probably not conventional notation but...
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