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Questions

Poducts z1 and z2 as a z1=5+3i and z2=4-2i, write the following in the form a+bi

In: Advanced Math

Rank of a Matrix Using the Echelon Form.

Find the Rank of a Matrix Using the Echelon Form of the above matrix.Give details step by step.

In: Advanced Math

Jordan Canonical Form

Let $ B_1 = \left\{(2,1,1,1),(1,1,1,1),(1,1,2,1)\right\} \hspace{2mm} $and $ \hspace{2mm}B_2 =\left\{(2,1,2,2)\right\}\hspace{2mm} $be two subsets of$ \hspace{2mm} \mathbb{R}^4, E_1 $ be a subspace spanned by$ B_1, E_2 $be a subspace spanned by $ B_2 $, and L be a linear operator on $ \mathbb{R}^4 $ defined by

$ L(v)=(-w +4x-y+z,-w+3x,-w+2x+y,-w+2x+z)\hspace{2mm},v=(w,x,y,z) $

(a) Show that $ B_1 $ is a basis for $ E_1 $ and $ B_2 $ is a basis for $ E_2 $

(b) Show that $ E_1 $ and $ E_2 $ are L-invariant. Find the matrices $ A_{1} =[L_{E_1}]_{B_1} $ and $ A_2=[L_{E_2}]_{B_2} $

In: Advanced Math

Jordan Canonical Form

Let A be a square matrix defined by $ A = \begin{pmatrix}6&2&3\\ -3&-1&-1\\ -5&-2&-2\end{pmatrix} $L be a map from$ \hspace{2mm} \mathbb{R}^3\hspace{2mm} $into$ \hspace{2mm}\mathbb{R}^3\hspace{2mm} $by$ \hspace{2mm} L(v) = Av. $

(a) Show that L is a linear operator on $ \hspace{2mm}\mathbb{R}^3. $

(b) Find the characteristic polynomial of L with respect to standard basis for $ \mathbb{R}^3 $ Derive the determinant of L then deduce that L is invertible.

(c) Find the eigenvalues and eigenspaces of L.

(d) Show that L is not diagonalizable, but it is triangularizable, then triangularize L.

(e) Write $ L^n $ in term of n, where $ L^n = L(L(...(L)..)) $, the n compositions of L.

In: Advanced Math

Jordan Fanonical Form

Let A be a square matrix defined by $ A =\begin{pmatrix}-1&-2&-1&3\\ -6&-5&1&6\\ -6&-4&0&6\\ -6&-7&1&8\end{pmatrix} $ and its characteristics polynomial $ P(\lambda)=\bigg(\lambda+1\bigg)^2\bigg(\lambda-2\bigg)^2 $

(a) Find the minimal polynomial of A.

(b) Deduce that A is not diagonalizable, but it is triangularizable, then triangularize A.

(c) Write $ A^n $ in terms of n.

In: Advanced Math

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} $

In: Advanced Math

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.

In: Advanced Math

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.

$ A=PDP^{-1},D=\begin{pmatrix}0&0&0\\ 0&-1-i&0\\ 0&0&-1+i\end{pmatrix},P=\begin{pmatrix}-1&-1-2i&-1+2i\\ 0&1+3i&1-3i\\ 1&1&1\end{pmatrix} $

In: Advanced Math

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.

In: Advanced Math

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.

In: Advanced Math

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 $ in term of n.

In: Advanced Math

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.

In: Advanced Math

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.

In: Advanced Math

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|\: $

In: Advanced Math

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.

In: Advanced Math