Matrix Exponentials: A, Eigenvalues, Jordan Form, Fundamental Matrix T

  • Thread starter Somefantastik
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In summary, you can find the eigenvalues of a matrix using the Jordan form, and using the Taylor series expansion, you can find the eigenvalues and derivatives of a matrix.
  • #1
Somefantastik
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[tex] A = \[ \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\] [/tex]

eigenvalues are [tex] \lambda_{1} = -1, \ \lambda_{2} = \lambda_{3} = 1 [/tex]

[tex](A-\lambda_{1}I)u^{(1)} = 0 \ => \ u^{(1)} = \[ \left( \begin{array}{c}
-1 \\
0 \\
1 \end{array} \right)\] [/tex]

[tex] (A-\lambda_{2}I)u^{(2)} = 0 \ => u^{(1)} = \[ \left( \begin{array}{c}
1 \\
0 \\
1 \end{array} \right)\] [/tex]

[tex] (A -\lambda_{3}I)u^{(3)} = u^{(2)} \ => \
\left(
\begin{array}{ccc|c}
1&0&-1&-1\\
0&0 &0&1\\
0&0&0&0
\end{array}
\right)
[/tex]

since we cannot have 0 = 1, we can say that there is only one eigenvector for

[tex]\lambda = 1 [/tex]

which means that the Jordan form will be

[tex] \[ \left( \begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1 \end{array} \right)\] [/tex]

Am I correct?

Now I need to find exp(Jt) and I'm not sure how.

If I only have 2 eigenvectors, how can I find the fund. matrix T such that

[tex]e^{At} = Te^{Jt}T^{-1} [/tex]?
 
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  • #2
In general, to find such a function of other than a "number", use the Taylor's series

eA= I+ A+ (1/2)A2+ (1/3!)A3+ ...

If A happens to be diagonal, that's easy: An is just the diagonal matrix with exponentials of the diagonal elements of A on its diagonal.

For Jordan form, its a little more complicated. Try calculating J2, J3, ... for this particular J yourself and see if you spot a pattern.
 
  • #3
I noticed that A2k = I and A(2k+1)'s = A, but I don't know how to get eA from that.
 
  • #4
Somefantastik said:
I noticed that A2k = I and A(2k+1)'s = A, but I don't know how to get eA from that.
How exactly are you having trouble summing the series?
 
  • #5
I'm just an idiot. I didn't realize that the Taylor expansions' limits came out to be cosh and sinh. I think I finished this problem. Thanks for the help so far.
 
  • #6
For the record, there's another method that can be useful. Note that you can separate that Jordan block into

[tex]
\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right]
=
\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]
+
\left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]
[/tex]

...
 
  • #7
Ah, I see, and the second matrix is nilpotent.

But still, if I got that, how would I compute

[tex]
e^{At} = Te^{Jt}T^{-1}
[/tex]

Not sure what T would be in this case.
 
  • #8
Right. And you'll find the results of doing arithmetic look a lot like derivatives -- a nilpotent matrix behaves a little bit like an infinitessimal.

(Again, look to the series to get started)
 

1. What is a matrix exponential?

A matrix exponential is a special type of function that can be applied to a matrix. It is defined as the sum of the infinite series of powers of the matrix, with each term divided by the factorial of its power.

2. What is the significance of eigenvalues in matrix exponentials?

Eigenvalues play a crucial role in calculating matrix exponentials. They are the values that, when multiplied by the identity matrix, give back the original matrix. In matrix exponentials, eigenvalues are used to raise the matrix to a certain power, making it easier to calculate.

3. What is the Jordan form of a matrix?

The Jordan form of a matrix is a special type of matrix that is used to simplify the calculation of matrix exponentials. It is a block diagonal matrix with Jordan blocks, which are square matrices with a particular form. The Jordan form of a matrix is unique up to the order of the blocks.

4. How is the fundamental matrix used in matrix exponentials?

The fundamental matrix is used to solve systems of linear differential equations. In matrix exponentials, it is used to find the solution to differential equations involving matrices. The fundamental matrix contains the eigenvectors and eigenvalues of the matrix, which are crucial in calculating matrix exponentials.

5. Can matrix exponentials be applied to non-square matrices?

No, matrix exponentials can only be applied to square matrices. This is because the definition of a matrix exponential involves raising the matrix to a power, which is only possible for square matrices. Non-square matrices do not have a well-defined exponential function.

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