Question

In: Advanced Math

Jordan Canonical Form

Let A be a square matrix defined by A=(323−2) 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 An A^n in term of n.

Solutions

Expert Solution

Solution

(a) Find the characteristic polynomial of A.

  ⟹  P(λ)=∣A−λI∣=λ2+λ−12=(λ−4)(λ+3λ) \implies P(\lambda)=|A-\lambda I|=\lambda^2+\lambda-12=\bigg(\lambda-4\bigg)\bigg(\lambda+3\lambda\bigg)

Therefore, the characteristics polynomial is 

P(λ)=∣A−λI∣=(λ−4)(λ+3λ) P(\lambda)=|A-\lambda I|=\bigg(\lambda-4\bigg)\bigg(\lambda+3\lambda\bigg)

(b) Show that A is diagonalizable then diagonalize it.

since, P(λ)splitted  ⟹  A P(\lambda)\hspace{2mm}splitted \implies A is diagonalizable.

∙Forλ1=4 \bullet\hspace{2mm} For \hspace{2mm}\lambda_1=4   ⟹  E4=span{(21)} \implies E_4=span\left\{\begin{pmatrix}2\\ 1\end{pmatrix}\right\}

∙Forλ2=−3 \bullet\hspace{2mm} For\hspace{2mm}\lambda_2=-3   ⟹  E−3=span{(−13)} \implies E_{-3}=span\left\{\begin{pmatrix}-1\\ 3\end{pmatrix}\right\}

Therefore, A=PDP−1 A=PDP^{-1} where

P=(2−313),D=(400−3),P−1=(3717−1727) P=\begin{pmatrix}2&-3\\ 1&3\end{pmatrix} ,D=\begin{pmatrix}4&0\\ 0&-3\end{pmatrix} ,P^{-1}=\begin{pmatrix}\frac{3}{7}&\frac{1}{7}\\ -\frac{1}{7}&\frac{2}{7}\end{pmatrix}

(c) Write An A^n in term of n.

since. A=PDP−1  ⟹  An=(PDP−1)n=PDnP−1 A=PDP^{-1} \implies A^n=\bigg(PDP^{-1}\bigg)^n=PD^nP^{-1}

  ⟹  An=(2−313)(4n00(−3)n)(3717−1727) \implies A^n=\begin{pmatrix}2&-3\\ 1&3\end{pmatrix}\begin{pmatrix}4^n&0\\ 0&\left(-3\right)^n\end{pmatrix}\begin{pmatrix}\frac{3}{7}&\frac{1}{7}\\ -\frac{1}{7}&\frac{2}{7}\end{pmatrix}

                =(3⋅ 22n+1+3(−3)n722n+1−6(−3)n73⋅ 4n−3(−3)n74n+6(−3)n7) =\begin{pmatrix}\frac{3\cdot \:2^{2n+1}+3\left(-3\right)^n}{7}&\frac{2^{2n+1}-6\left(-3\right)^n}{7}\\ \frac{3\cdot \:4^n-3\left(-3\right)^n}{7}&\frac{4^n+6\left(-3\right)^n}{7}\end{pmatrix}

Therefore, An=(3⋅ 22n+1+3(−3)n722n+1−6(−3)n73⋅ 4n−3(−3)n74n+6(−3)n7) A^n=\begin{pmatrix}\frac{3\cdot \:2^{2n+1}+3\left(-3\right)^n}{7}&\frac{2^{2n+1}-6\left(-3\right)^n}{7}\\ \frac{3\cdot \:4^n-3\left(-3\right)^n}{7}&\frac{4^n+6\left(-3\right)^n}{7}\end{pmatrix}

 

Answer

Therefore. 

a). P(λ)=∣A−λI∣=(λ−4)(λ+3λ) P(\lambda)=|A-\lambda I|=\bigg(\lambda-4\bigg)\bigg(\lambda+3\lambda\bigg)

b). A=PDP−1 A=PDP^{-1} where P=(2−313),D=(400−3) P=\begin{pmatrix}2&-3\\ 1&3\end{pmatrix} ,D=\begin{pmatrix}4&0\\ 0&-3\end{pmatrix} ,P−1=(3717−1727) P^{-1}=\begin{pmatrix}\frac{3}{7}&\frac{1}{7}\\ -\frac{1}{7}&\frac{2}{7}\end{pmatrix}

c). An=(3⋅ 22n+1+3(−3)n722n+1−6(−3)n73⋅ 4n−3(−3)n74n+6(−3)n7) A^n=\begin{pmatrix}\frac{3\cdot \:2^{2n+1}+3\left(-3\right)^n}{7}&\frac{2^{2n+1}-6\left(-3\right)^n}{7}\\ \frac{3\cdot \:4^n-3\left(-3\right)^n}{7}&\frac{4^n+6\left(-3\right)^n}{7}\end{pmatrix}

SUN BUNRA answered 2 years ago
Next > < Previous

Related Solutions

Jordan Canonical Form
Let A∈Mn(R) A\in M_n(\mathbb{R})\hspace{2mm} and λ1,λ2,...,λn \hspace{2mm}\lambda_1, \lambda_2,...,\lambda_n \hspace{2mm} (no need distinct) be eigenvalues of A. Show that  a). ∑i=1nλi=tr(A) \sum _{i=1}^n\lambda _i=tr\left(A\right)   b).  ∏i=1nλi=∣A∣  \:\prod _{i=1}^n\lambda _i=\left|A\right|\:  
Jordan Canonical Form
Let A∈Mn(R) A\in M_n(\mathbb{R})\hspace{2mm} andmA(λ) \hspace{2mm} m_A(\lambda)\hspace{2mm} be its minimal polynomial. Let f be a polynomial satisfiesf(A)=0. \hspace{2mm}f(A) = 0. \hspace{2mm} Show thatf(λ) \hspace{2mm} f(\lambda) \hspace{2mm} is divisible bymA(λ). \hspace{2mm} m_A(\lambda).
Jordan Canonical Form
Let A be a square matrix defined by A=(4−212012−23) A=\begin{pmatrix}4&-2&1\\ 2&0&1\\ 2&-2&3\end{pmatrix}\hspace{2mm} Find the minimal polynomial of A. Then express A4 A^4 and A−1 A^{-1} in terms of A and I.
Jordan Canonical Form
Determine the value of a so that λ=2 \lambda = 2 is an eigenvalue of  A=(1−10a1101+a3) 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∈M6(R) A\in M_6(\mathbb{R}) be an invertible matrix satisfies A3−4A2+3A=0 A^3-4A^2 + 3A = 0 and tr(A)=8. tr(A) = 8. Find the characteristics polynomial of A.  
Jordan Canonical Form
Let A be a square matrix defined by A=(−3−1−3525−1−1−1) 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 R \mathbb{R} (c) Show that A is diagonalizable overC \mathbb{C} . Find the eigenspaces. Diagonalize A. (d) Express An A^n in the form of anA2+bnA+cnIn a_nA^2+b_ nA+c_nI_n where (an),(bn) (a_n), (b_n) and (cn) (c_n) are real sequences to be specified....
Jordan Canonical Form
Let A be a square matrix defined by A=(−2−1−5223422) 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 (a)_n, (b)_n ,(c)_n satisfying. {an+1=−2an−bn−5cn,a0=1bn+1=2an+2bn+3cn,b0=0cn+1=4an+2bn+6cn,c0=1 \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=(−8−3−6404422) 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 An A^n in terms of I,A,A2 I, A,A^2 and n.  
Jordan Canonical Form
Let A be a square matrix defined byA=(2−311−211−32) 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=(−13−1−35−1−331) 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 An A^n in term of n.
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT