Question

In: Advanced Math

Jordan Canonical Form

Let $A\in M_n(\mathbb{R})\hspace{2mm}$ and $\hspace{2mm} m_A(\lambda)\hspace{2mm}$ be its minimal polynomial. Let f be a polynomial satisfies $\hspace{2mm}f(A) = 0. \hspace{2mm}$ Show that $\hspace{2mm} f(\lambda) \hspace{2mm}$ is divisible by $\hspace{2mm} m_A(\lambda).$

Expert Solution

solution

we have $f(\lambda)=m_A(\lambda)g(\lambda)+r(\lambda) \hspace{3mm}(1)$

$deg\bigg(r(\lambda)\bigg)$ $<\deg$ $\bigg(m_A(\lambda)\bigg)$ $\hspace{4mm}(2)$

+Supposse that $r(\lambda)=0 \hspace{2mm}$ From (1). we have

$\implies f(A)=m_A(A)q(A)+r(A)$

Since. $f(A)=0$ and $m_A(A)=0$

$\implies r(A)=0 ,\hspace{2mm}$ with (2), contradiction to the definition of minimal polynomial. Then, $r(\lambda)=0$

Answer

Therefore $f(\lambda)$ is divisible by $m_A(\lambda).$

SUN BUNRA answered 2 years ago

Jordan Canonical Form

Let

A\in M_n(\mathbb{R})\hspace{2mm}

and

\hspace{2mm}\lambda_1, \lambda_2,...,\lambda_n \hspace{2mm}

(no need distinct) be eigenvalues of A. Show that a).

\sum _{i=1}^n\lambda _i=tr\left(A\right)

b).

\:\prod _{i=1}^n\lambda _i=\left|A\right|\:

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Let A be a square matrix defined by

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and

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in terms of A and I.

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Determine the value of a so that

\lambda = 2

is an eigenvalue of

A=\begin{pmatrix}1&-1&0\\ a&1&1\\ 0&1+a&3\end{pmatrix}

Then show that A is diagonallizable and diagonalize it.

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Let

A\in M_6(\mathbb{R})

be an invertible matrix satisfies

A^3-4A^2 + 3A = 0

and

tr(A) = 8.

Find the characteristics polynomial of A.

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Let A be a square matrix defined by

A = \begin{pmatrix}-3&-1&-3\\ 5&2&5\\ -1&-1&-1\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Find the eigenvalues of A. Show that A is not diagonalizable over

\mathbb{R}

(c) Show that A is diagonalizable over

\mathbb{C}

. Find the eigenspaces. Diagonalize A. (d) Express

A^n

in the form of

a_nA^2+b_ nA+c_nI_n

where

(a_n), (b_n)

and

(c_n)

are real sequences to be specified....

Jordan Canonical Form

Let A be a square matrix defined by

A = \begin{pmatrix}-2&-1&-5\\ 2&2&3\\ 4&2&2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Find the eigenvalues and eigenspaces of A. (c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A. (d) Find the three real sequences

(a)_n, (b)_n ,(c)_n

satisfying.

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Jordan Canonical Form

Let A be a square matrix defined by

A =\begin{pmatrix}-8&-3&-6\\ 4&0&4\\ 4&2&2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Find the eigenvalues and eigenspaces of A. (c) Show that A is not diagonalizable, but it is triangularizable, then triangularize A. (d) Write

A^n

in terms of

I, A,A^2

and n.

Jordan Canonical Form

Let A be a square matrix defined by

A =\begin{pmatrix}2&-3&1\\ 1&-2&1\\ 1&-3&2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write $A^n$ \hspace{2mm} in term of n.

Jordan Canonical Form

Let A be a square matrix defined by

A =\begin{pmatrix}-1&3&-1\\ -3&5&-1\\ -3&3&1\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write

A^n

in term of n.

Jordan Canonical Form

Let A be a square matrix defined by

A = \begin{pmatrix}3&2\\ 3&-2\end{pmatrix}

(a) Find the characteristic polynomial of A. (b) Show that A is diagonalizable then diagonalize it. (c) Write

A^n

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