Intereting Posts

Integral $\int_0^\infty\sin{(x^4)} dx$
Classify sphere bundles over a sphere
Evaluating $\int_0^\infty\frac{\log^{10} x}{1 +x^3}dx$
If $0<a<1, 0<b<1$, $a+b=1$, then prove that $a^{2b}+ b^{2a} \le 1$
Field reductions. part two
True or false or not-defined statements
Why do I get two different results for the reciprocal of $i$?
Finding the area of a implicit relation
Clearing gap in Munkres
Compute $\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$
For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero
Intuition behind arc length formula
Learning Fibre Bundle from “Topology and Geometry” by Bredon
Understanding the idea of a Limit Point (Topology)
Recognizing that a function has no elementary antiderivative

If I have a differential equation $ y'(t)=A y(t)$ where A is a constant square matrix that is not diagonalizable(although it is surely possible to calculate the eigenvalues) and no initial condition is given. And now I am interested in the fundamental matrix. Is there a general method to determine this matrix? I do not want to use the exponential function and the Jordan normal form, as this is quite exhausting. Maybe there is also an ansatz possible as it is for the special case, where this differential equation is equivalent to an n-th order ode.

I saw a method where they calculated the eigenvalues of the matrix and depending on the multiplicity n of this eigenvalue they used an exponential term(with the eigenvalue) and in each component an n-th order polynomial as a possible ansatz. Though they only did this, when they were interested in a initial value problem, so with an initial condition and not for a general solution.

I was asked to deliver an example: so $y'(t)=\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} y(t)$ If somebody can construct a fundamental matrix for this system, than this should be sufficient

- Challenge: Demonstrate a Contradiction in Leibniz' differential notation
- Dedekind's cut and axioms
- Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$
- Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?
- Why Cauchy's definition of infinitesimal is not widely used?
- Find the value of : $\lim\limits_{n\to \infty} \sqrt {\frac{(3n)!}{n!(2n+1)!}} $

- Applying trigonometry in solving quintic polynomials?
- Find the Vector Equation of a line perpendicular to the plane.
- Deriving even odd function expressions
- What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?
- Second-order non-linear ODE
- Fitzpatrick's proof of Darboux sum comparison lemma
- What is the limit of the following sum
- Thickness of the Boundary Layer
- Injection and Bijection of the function $f(x,y)=(\frac{x}{1+x+y},\frac{y}{1+x+y}).$
- limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$

We have many ways to proceed and this is only a $2×2$. We can choose from:

*Nineteen Dubious Ways to Compute the Exponential of a*

Matrix, Twenty-Five Years Later- Putzer’s Method 1 and Method 2
- For non-repeated eigenvalues, we can simply write:

$$ x(t) = e^{At}x_0 = Pe^{Jt}P^{-1}x_0 = c_1v_1e^{\lambda_1 t} + \ldots + c_nv_ne^{\lambda_n t} $$ - The Direct Method for repeated eigenvalues

$$\tag 1 e^{At} = \left[I+ \sum_{k=1}^\infty \dfrac{(A-\lambda I)^k}{k!}t^k\right]e^{\lambda t}$$

For the matrix $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$, we have:

$\det [A-\lambda I] = \det \begin{bmatrix} 3-\lambda & -4 \\ 1 & -1-\lambda \end{bmatrix} = 0 \rightarrow \lambda^2-2 \lambda+1 = 0 \rightarrow \lambda_{1,2} = 1,1$ (a double eigenvalue). From the eigenvalues, we derive the eigenvalue/eigenvector pairs:

- $\lambda_1 = 1, v_1 = (2, 1)$
- $\lambda_2 = 1, v_2 = (1, 0)$ (the second eigenvector is a generalized one)

Lets find the matrix exponential using two different methods.

**Method 1**

From $(1)$, we have:

$$e^{At} = \left[I + \dfrac{(A-\lambda I)^0}{1!}t^1 \right]e^{\lambda t} = \left[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 2 & -4 \\ 1 & -2 \end{bmatrix}t\right] = e^{t}\begin{bmatrix} 1+2t & -4t \\ t & 1-2t \end{bmatrix} $$

**Method 2**

Use the Laplace Transform.

$$e^{At}=\mathcal{L}^{-1}\left((sI-A)^{-1}\right)=

\mathcal{L}^{-1}\left(\begin{bmatrix}s-3 & 4 \\ -1 & s+1\end{bmatrix}^{-1}\right)

= e^{t}\begin{bmatrix}1+2 t & -4 t \\ t & 1-2 t\end{bmatrix}$$

$\vdots$

**Method n**

Try other approaches discussed above!

**Update: Method n+1**

If you wanted to write $A$ using *Jordan Normal Form*, we would have:

$$A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} = PJP^{-1} = \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \\1 & -2 \end{bmatrix}$$

To write the matrix exponential for this, we take advantage of the Jordan Block and have:

$$e^{At} = e^{PJP^{-1}t} = Pe^{Jt}P^{-1} = \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} \cdot e^{\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}t} \cdot \begin{bmatrix} 0 & 1 \\1 & -2 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} e^t & te^t \\ 0 & e^t\end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \\1 & -2 \end{bmatrix} = e^{t}\begin{bmatrix}1+2 t & -4 t \\ t & 1-2 t\end{bmatrix}$$

Lastly, it is worth noting that sometimes the Fundamental Matrix is given as:

$$\phi(t, t_0) = \phi(t) \cdot \phi^{-1}(t_0)$$

There are many methods for determining the matrix exponential, even for a non-diagonalizable matrix. One of the easiest is via the Laplace transform. You can check that

$$\mathcal{L}(e^{tA})(s) = (sI-A)^{-1}.$$

For your example,

$$\mathcal{L}(e^{tA})(s) = \begin{bmatrix}s-3 & 4 \\ -1 & s+1\end{bmatrix}^{-1} = \frac{1}{(s-1)^2}\begin{bmatrix}s+1 & -4 \\ 1 & s-3\end{bmatrix},$$

and a (component-wise) inverse Laplace transform gives

$$

e^{tA} = \begin{bmatrix}(2t+1)e^t & -4te^t \\ te^t & -(2t-1)e^t\end{bmatrix}.

$$

Formally, this is only valid for $t>0$, but since the elements in $e^{tA}$ are holomorphic, the identity theorem for holomorphic functions shows that the equality is valid for all $t$.

- The magnitude of the difference between the integral and the Riemann sums for continuous functions
- Mean value theorem for the second derivative, when the first derivative is zero at endpoints
- f(a) = inverse of a is an isomorphism iff a group G is Abelian
- Expected number of cards in the stack?
- Why do you add +1 in counting test questions?
- Does $\lim_{(x,y) \to (0,0)} \frac{x^4+y^4}{x^3+y^3}$ exist?
- Estimation of integral
- Proving all rational numbers including negatives are countable
- Expected Value with stopping rule.
- Proving a simple graph is a connected graph
- Cardinality of the set of all real functions of real variable
- Independence and Events.
- Can we found mathematics without evaluation or membership?
- Looking for help with this elementary method of finding integer solutions on an elliptic curve.
- Interpretation for the determinant of a stochastic matrix?