Find a characteristic polynomial of endomorphism $varphi$The characteristic and minimal polynomial of a...

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Find a characteristic polynomial of endomorphism $varphi$


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Let $ A = (alpha_ {1}, ..., alpha_ {n}) $ be a basis of a vector space $ V $ and let $ varphi: V rightarrow V $ be the endomorphism given by the conditions $varphi(alpha _{i})=alpha _{i+1}$ for $i=1,...,n-1$ and $varphi(alpha _{n})=a_{0} alpha_{1}+a_{1} alpha_{2}+...+a_{n-1} alpha_{n}$. Find a characteristic polynomial of endomorphism $varphi$.




In this task I have a matrix $M^{A}_{A}$ which has $alpha_{i+1}$ in subsequent columns and this $alpha_{i+1}$. However I completely don't know how to do this task because this fact is insufficient for me.

Can you help me?






linear-algebra matrices determinant







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
    $endgroup$
    – darij grinberg
    yesterday
















2












$begingroup$



Let $ A = (alpha_ {1}, ..., alpha_ {n}) $ be a basis of a vector space $ V $ and let $ varphi: V rightarrow V $ be the endomorphism given by the conditions $varphi(alpha _{i})=alpha _{i+1}$ for $i=1,...,n-1$ and $varphi(alpha _{n})=a_{0} alpha_{1}+a_{1} alpha_{2}+...+a_{n-1} alpha_{n}$. Find a characteristic polynomial of endomorphism $varphi$.




In this task I have a matrix $M^{A}_{A}$ which has $alpha_{i+1}$ in subsequent columns and this $alpha_{i+1}$. However I completely don't know how to do this task because this fact is insufficient for me.

Can you help me?






linear-algebra matrices determinant







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
    $endgroup$
    – darij grinberg
    yesterday














2












2








2


1



$begingroup$



Let $ A = (alpha_ {1}, ..., alpha_ {n}) $ be a basis of a vector space $ V $ and let $ varphi: V rightarrow V $ be the endomorphism given by the conditions $varphi(alpha _{i})=alpha _{i+1}$ for $i=1,...,n-1$ and $varphi(alpha _{n})=a_{0} alpha_{1}+a_{1} alpha_{2}+...+a_{n-1} alpha_{n}$. Find a characteristic polynomial of endomorphism $varphi$.




In this task I have a matrix $M^{A}_{A}$ which has $alpha_{i+1}$ in subsequent columns and this $alpha_{i+1}$. However I completely don't know how to do this task because this fact is insufficient for me.

Can you help me?






linear-algebra matrices determinant







share|cite|improve this question











$endgroup$





Let $ A = (alpha_ {1}, ..., alpha_ {n}) $ be a basis of a vector space $ V $ and let $ varphi: V rightarrow V $ be the endomorphism given by the conditions $varphi(alpha _{i})=alpha _{i+1}$ for $i=1,...,n-1$ and $varphi(alpha _{n})=a_{0} alpha_{1}+a_{1} alpha_{2}+...+a_{n-1} alpha_{n}$. Find a characteristic polynomial of endomorphism $varphi$.




In this task I have a matrix $M^{A}_{A}$ which has $alpha_{i+1}$ in subsequent columns and this $alpha_{i+1}$. However I completely don't know how to do this task because this fact is insufficient for me.

Can you help me?






linear-algebra matrices determinant




linear-algebra matrices determinant






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









Bernard

122k741116




122k741116










asked yesterday









MP3129MP3129

47910




47910












  • $begingroup$
    Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
    $endgroup$
    – darij grinberg
    yesterday


















  • $begingroup$
    Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
    $endgroup$
    – darij grinberg
    yesterday
















$begingroup$
Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
$endgroup$
– darij grinberg
yesterday




$begingroup$
Your matrix $M_A^A$ is the companion matrix of a monic polynomial with non-leading coefficients $a_0, a_1, ldots, a_{n-1}$ (maybe reversed, maybe transposed, depending on how a companion matrix is defined). As for computing the characteristic polynomial of a companion matrix, see math.stackexchange.com/questions/10216 (although the asker of that question has excluded the simplest answer).
$endgroup$
– darij grinberg
yesterday










1 Answer
1






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oldest

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$begingroup$

Hint:
The matrix of this endomorphism is
$$begin{bmatrix}
0&0&0&dots&0&a_0\
1&0&0&dots&0&a_1\
0&1&0&dots &0&a_2 \[-1ex]
vdots&vdots&ddots&&vdots&vdots\
0&0&0&dots&0&a_{n-2}\
0&0&0&dots&1&a_{n-1}
end{bmatrix},$$
so its characteristic polynomial is the determinant
$$begin{vmatrix}
-x&0&0&dots&0&a_0\
1&-x&0&dots&0&a_1\
0&1&-x&dots &0&a_2 \[-1ex]
vdots&vdots&ddots&&vdots&vdots\
0&0&0&dots&-x&a_{n-2}\
0&0&0&dots&1&a_{n-1}-x
end{vmatrix}.$$
Denote it as $D(a_0,a_1,dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation:
$$D(a_0,a_1,dots,a_{n-1})=-xD(a_1,dots,a_{n-1})+(-1)^{n-1}a_0.$$
Deduce from this relation that
$$D(a_0,a_1,dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-dots-a_1x-a_0).$$






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    1 Answer
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    1 Answer
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    active

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    2












    $begingroup$

    Hint:
    The matrix of this endomorphism is
    $$begin{bmatrix}
    0&0&0&dots&0&a_0\
    1&0&0&dots&0&a_1\
    0&1&0&dots &0&a_2 \[-1ex]
    vdots&vdots&ddots&&vdots&vdots\
    0&0&0&dots&0&a_{n-2}\
    0&0&0&dots&1&a_{n-1}
    end{bmatrix},$$
    so its characteristic polynomial is the determinant
    $$begin{vmatrix}
    -x&0&0&dots&0&a_0\
    1&-x&0&dots&0&a_1\
    0&1&-x&dots &0&a_2 \[-1ex]
    vdots&vdots&ddots&&vdots&vdots\
    0&0&0&dots&-x&a_{n-2}\
    0&0&0&dots&1&a_{n-1}-x
    end{vmatrix}.$$
    Denote it as $D(a_0,a_1,dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation:
    $$D(a_0,a_1,dots,a_{n-1})=-xD(a_1,dots,a_{n-1})+(-1)^{n-1}a_0.$$
    Deduce from this relation that
    $$D(a_0,a_1,dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-dots-a_1x-a_0).$$






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Hint:
      The matrix of this endomorphism is
      $$begin{bmatrix}
      0&0&0&dots&0&a_0\
      1&0&0&dots&0&a_1\
      0&1&0&dots &0&a_2 \[-1ex]
      vdots&vdots&ddots&&vdots&vdots\
      0&0&0&dots&0&a_{n-2}\
      0&0&0&dots&1&a_{n-1}
      end{bmatrix},$$
      so its characteristic polynomial is the determinant
      $$begin{vmatrix}
      -x&0&0&dots&0&a_0\
      1&-x&0&dots&0&a_1\
      0&1&-x&dots &0&a_2 \[-1ex]
      vdots&vdots&ddots&&vdots&vdots\
      0&0&0&dots&-x&a_{n-2}\
      0&0&0&dots&1&a_{n-1}-x
      end{vmatrix}.$$
      Denote it as $D(a_0,a_1,dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation:
      $$D(a_0,a_1,dots,a_{n-1})=-xD(a_1,dots,a_{n-1})+(-1)^{n-1}a_0.$$
      Deduce from this relation that
      $$D(a_0,a_1,dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-dots-a_1x-a_0).$$






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Hint:
        The matrix of this endomorphism is
        $$begin{bmatrix}
        0&0&0&dots&0&a_0\
        1&0&0&dots&0&a_1\
        0&1&0&dots &0&a_2 \[-1ex]
        vdots&vdots&ddots&&vdots&vdots\
        0&0&0&dots&0&a_{n-2}\
        0&0&0&dots&1&a_{n-1}
        end{bmatrix},$$
        so its characteristic polynomial is the determinant
        $$begin{vmatrix}
        -x&0&0&dots&0&a_0\
        1&-x&0&dots&0&a_1\
        0&1&-x&dots &0&a_2 \[-1ex]
        vdots&vdots&ddots&&vdots&vdots\
        0&0&0&dots&-x&a_{n-2}\
        0&0&0&dots&1&a_{n-1}-x
        end{vmatrix}.$$
        Denote it as $D(a_0,a_1,dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation:
        $$D(a_0,a_1,dots,a_{n-1})=-xD(a_1,dots,a_{n-1})+(-1)^{n-1}a_0.$$
        Deduce from this relation that
        $$D(a_0,a_1,dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-dots-a_1x-a_0).$$






        share|cite|improve this answer









        $endgroup$



        Hint:
        The matrix of this endomorphism is
        $$begin{bmatrix}
        0&0&0&dots&0&a_0\
        1&0&0&dots&0&a_1\
        0&1&0&dots &0&a_2 \[-1ex]
        vdots&vdots&ddots&&vdots&vdots\
        0&0&0&dots&0&a_{n-2}\
        0&0&0&dots&1&a_{n-1}
        end{bmatrix},$$
        so its characteristic polynomial is the determinant
        $$begin{vmatrix}
        -x&0&0&dots&0&a_0\
        1&-x&0&dots&0&a_1\
        0&1&-x&dots &0&a_2 \[-1ex]
        vdots&vdots&ddots&&vdots&vdots\
        0&0&0&dots&-x&a_{n-2}\
        0&0&0&dots&1&a_{n-1}-x
        end{vmatrix}.$$
        Denote it as $D(a_0,a_1,dots,a_{n-1})$ and, expanding along the first row, prove the recurrence relation:
        $$D(a_0,a_1,dots,a_{n-1})=-xD(a_1,dots,a_{n-1})+(-1)^{n-1}a_0.$$
        Deduce from this relation that
        $$D(a_0,a_1,dots,a_{n-1})=(-1)^n(x^n-a_{n-1}x^{n-1}-dots-a_1x-a_0).$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        BernardBernard

        122k741116




        122k741116






























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