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Minimal polynomial: Let $A$ be an $n \times n$ matrix. By definition, the minimal polynomial of $A$ is the monic polynomial $m_A(t)=t^k+c_{k-1} t^{k-1}+\cdots+c_1 t+c_0$ of minimal degree $k$ that annihilates $A$, so $m_A(A)=A^k+c_{k-1} A^{k-1}+\cdots+c_1 A+c_0 \mathrm{I}=0$. (a) Prove that the monic minimal polynomial $m_A$ is unique. (b) (c) Prove that if $r(t)$ is any other polynomial such that $r(A)=0$, then $r(t)=q(t) m_A(t)$ for some polynomial $q(t)$. (d) Prove that the matrix's minimal polynomial is a factor of its characteristic polynomial, so $p_A(t)=q_A(t) m_A(t)$ for some polynomial $q_A(t)$. Hint: Use the Cayley-Hamilton Theorem in Exercise 8.6.22. (e) Prove that if $A$ has all distinct eigenvalues, then $p_A=m_A$. (f) Prove that $p_A=m_A$ if and only if no two Jordan blocks have the same eigenvalue.

    Minimal polynomial: Let $A$ be an $n \times n$ matrix. By definition, the minimal polynomial of $A$ is the monic polynomial $m_A(t)=t^k+c_{k-1} t^{k-1}+\cdots+c_1 t+c_0$ of minimal degree $k$ that annihilates $A$, so $m_A(A)=A^k+c_{k-1} A^{k-1}+\cdots+c_1 A+c_0 \mathrm{I}=0$. (a) Prove that the monic minimal polynomial $m_A$ is unique. (b) (c) Prove that if $r(t)$ is any other polynomial such that $r(A)=0$, then $r(t)=q(t) m_A(t)$ for some polynomial $q(t)$. (d) Prove that the matrix's minimal polynomial is a factor of its characteristic polynomial, so $p_A(t)=q_A(t) m_A(t)$ for some polynomial $q_A(t)$. Hint: Use the Cayley-Hamilton Theorem in Exercise 8.6.22. (e) Prove that if $A$ has all distinct eigenvalues, then $p_A=m_A$. (f) Prove that $p_A=m_A$ if and only if no two Jordan blocks have the same eigenvalue.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 8, Problem 23

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Step 1

- Consider the polynomial $d(t) = m_A(t) - n_A(t)$. Since both $m_A(t)$ and $n_A(t)$ are monic and of the same degree, $d(t)$ has a degree less than $k$. - Since $m_A(A) = 0$ and $n_A(A) = 0$, it follows that $d(A) = m_A(A) - n_A(A) = 0 - 0 = 0$. - If Show more…

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Minimal polynomial: Let $A$ be an $n \times n$ matrix. By definition, the minimal polynomial of $A$ is the monic polynomial $m_A(t)=t^k+c_{k-1} t^{k-1}+\cdots+c_1 t+c_0$ of minimal degree $k$ that annihilates $A$, so $m_A(A)=A^k+c_{k-1} A^{k-1}+\cdots+c_1 A+c_0 \mathrm{I}=0$. (a) Prove that the monic minimal polynomial $m_A$ is unique. (b) (c) Prove that if $r(t)$ is any other polynomial such that $r(A)=0$, then $r(t)=q(t) m_A(t)$ for some polynomial $q(t)$. (d) Prove that the matrix's minimal polynomial is a factor of its characteristic polynomial, so $p_A(t)=q_A(t) m_A(t)$ for some polynomial $q_A(t)$. Hint: Use the Cayley-Hamilton Theorem in Exercise 8.6.22. (e) Prove that if $A$ has all distinct eigenvalues, then $p_A=m_A$. (f) Prove that $p_A=m_A$ if and only if no two Jordan blocks have the same eigenvalue.

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