Suppose A^2= A , where A is an n by n matrix. Use Jordan canonical form...

Suppose A^2= A , where A is an n by n matrix. Use Jordan canonical form to thaw that A is diagonalizable.

Expert Solution

Colby Messinger 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|\: $

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). $

Jordan Canonical Form

Let A be a square matrix defined by $ A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix}\hspace{2mm} $Find the minimal polynomial of A. Then express $ A^4 $ and $ A^{-1} $ in terms of A and I.

Jordan Canonical Form

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.

Jordan Canonical Form

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.

Jordan Canonical Form

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. $ \begin{cases} a_{n+1}=-2a_n-b_n-5c_n \hspace{2mm},a_0=1 & \quad \\ b_{n+1}=2a_n+2b_n+3c_n \hspace{2mm}, b_0=0 & \quad \\ c_{n+1}=4a_n+2b_n+6c_n \hspace{2mm},c_0=1 & \quad \end{cases} $

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.

Question

Suppose A^2= A , where A is an n by n matrix. Use Jordan canonical form...

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Jordan Canonical Form

Jordan Canonical Form

Jordan Canonical Form

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Jordan Canonical Form

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Jordan Canonical Form

Jordan Canonical Form