MHB Matrix Exponential: Find Jordan Form & Compute eA

  • Thread starter Thread starter Fernando Revilla
  • Start date Start date
  • Tags Tags
    Exponential Matrix
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
I quote a question from Yahoo! Answers

Consider the following Matrix:
A =
[ 1 -1 0
1 3 0
4 6 -1 ]

(a) Find a Jordan form of the matrix, as well as a basis that corresponds to that Jordan form.
(b) Compute the exponential matrix eA.

I have given a link to the topic there so the OP can see my response.
 
Mathematics news on Phys.org
$(a)$ The eigenvalues of $A=\begin{bmatrix}{1}&{-1}&{\;\;0}\\{1}&{\;\;3}&{\;\;0}\\{4}&{\;\;6}&{-1}\end{bmatrix}$ are: $$\det (A-\lambda I)=(-1-\lambda)(\lambda -2)^2=0\Leftrightarrow \lambda=-1\mbox{ (simple) }\vee \;\lambda=2\mbox{ (double)}$$ We have $\dim\ker (A+I)=1$ (simple eigenvalue) and $\dim \ker (A-2I)=3-\mbox{rank} (A-2I)=3-2=1$. So the canonical form of Jordan is $$J= \begin{bmatrix} {-1}&{0}&{0}\\{0}&{2}&{1}\\{0}&{0}&{2}\end{bmatrix} $$ A basis for $\ker (A+I)$ is $\{v=(0,0,1)^T\}$. Now, we need two linearly independent vectors $e_1,e_2$ such that $(A-2I)e_1=0$ and $(A-2I)e_2=e_1$. We easily find $e_1=(-3,3,2)^T$ and $e_2=(8,-5,0)^T$. As a consequence, the transition matrix $P$ satisfying $P^{-1}AP=J$ is $$P=[v\;\;e_1\;\;e_2]=\begin{bmatrix}{0}&{-3}&{\;\;8}\\{0}&{\;\;3}&{-5}\\{1}&{\;\;2}&{\;\;0}\end{bmatrix}$$

$(b)\;\;e^{A}=Pe^{J}P^{-1}=P\;\begin{bmatrix}{e^{-1}}&{0}&{0}\\{0}&{e^{2}}&{e^{2}}\\{0}&{0}&{e^{2}} \end{bmatrix}\;P^{-1}=\ldots=\begin{bmatrix}{e}&{e^{-1}}&{1}\\{e}&{e^3}&{1}\\{e^4}&{e^6}&{e^{-1}} \end{bmatrix}$
 
Last edited:
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Back
Top