Question regarding the exponential of a matrix

In summary, the conversation discusses solving a question involving matrices and exponentials. The problem is related to the top-right element of the matrix when summed, as it results in negative values of n. The conversation suggests using the original multiplication with n before cancelling it out with the denominator to get rid of the n=0 term. It is also mentioned that the solution can be found by using the binomial formula and splitting the exponential into two parts.
  • #1
ChrisVer
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Homework Statement
Show that you can write [itex]e^{\textbf{A}t} = f_1(t) \textbf{1} + f_2(t) \textbf{A} [/itex] and determine [itex]f_{1},~f_{2}[/itex] if [itex]\textbf{A}= \begin{pmatrix} a & b \\ 0 & a \end{pmatrix}[/itex]
Relevant Equations
[itex]e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} [/itex]
Hi, I think this is a nitpicking question, but oh well let me hear your inputs.
Actually I tried to solve this question straightforwardly, by Taylor expanding the exponential and showing that:
[itex] \textbf{A}^n = \begin{pmatrix} a^n & nba^{n-1} \\ 0 & a^n \end{pmatrix} [/itex]
i.e.
[itex]e^{\textbf{A}t} = \sum_{n=0}^{\infty} \frac{\textbf{A}^n t^n}{n!}= \begin{pmatrix} \sum_{n=0}^{\infty} \frac{a^nt^n}{n!} & bt \sum_{n=0}^{\infty} \frac{na^{n-1} t^{n-1}}{n!} \\ 0 & \sum_{n=0}^{\infty} \frac{a^nt^n}{n!} \end{pmatrix} [/itex]
(forget [itex]t[/itex] for now)

The problem I faced was related to the top-right element of the matrix when summed. As the exponential gets expanded, it gives me a sum running from n=0 to infinity. However that term results to negative values of n when n=0:
[itex]\sum_{n=0}^{\infty} b n \frac{a^{n-1}}{n!} = b \sum_{n=0}^{\infty} \frac{a^{n-1}}{(n-1)!} [/itex]
Which doesn't make much sense. However, I don't know how I could emit the term with [itex]n=0[/itex]. I could try to say that [itex]m = n-1[/itex] meaning it would change to [itex] \sum_{m=-1}^{\infty} \frac{a^{m}}{m!} [/itex], but I have the exact same problem.

(I already know how the matrix is written and that term should be a [itex]be^{a}[/itex])
 
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  • #2
Why don't you write ##A=a\cdot I + b\cdot N##? Since ##N^2=0## and ##[I,N]=0## this should be the easiest way to write down ##(aI+bN)^n##.
 
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  • #3
Thanks. I think that what can actually help me get rid of the [itex]n=0[/itex] term is the original multiplication with [itex]n[/itex], before I cancel it out with the denominator...
 
  • #4
ChrisVer said:
Thanks. I think that what can actually help me get rid of the [itex]n=0[/itex] term is the original multiplication with [itex]n[/itex], before I cancel it out with the denominator...
That term should never be in there, s ##A^0 = I##, so the zeroth term is ##0## and has no terms in ##a## or ##b##.
 
  • #5
ChrisVer said:
[itex] \textbf{A}^n = \begin{pmatrix} a^n & nba^{n-1} \\ 0 & a^n \end{pmatrix} [/itex]

This is valid for ##n \ge 1##.
 
  • #6
PeroK said:
This is valid for ##n \ge 1##.
Hm, yes. But I think it is expandable to ##n=0## too, no? (Or it happens to be the case because of this "type" of matrix)
I say that because ##nba^{n-1} =0##, and so we obtain the unit matrix as expected. I agree that I was too fast to cancel out ##n## with ##n!##, and I missed that I could omit the sum for ##n=0## right away.
 
  • #7
ChrisVer said:
Hm, yes. But I think it is expandable to ##n=0## too, no? (Or it happens to be the case because of this "type" of matrix)
I say that because ##nba^{n-1} =0##, and so we obtain the unit matrix as expected. I agree that I was too fast to cancel out ##n## with ##n!##, and I missed that I could omit the sum for ##n=0## right away.
Okay, but you can drop the ##n = 0## term, otherwise, when you cancel ##n##, you are cancelling ##0##.
 
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  • #8
You should really calculate ## (aI+bN)^n ## by the binomial formula. With ##(...)^0=I## and ##N^2=0## you can almost write the answer without any calculation. Since ##[I,N]=0## you can also calculate ##e^{tA}=e^{atI}\cdot e^{tbN}##.
 
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  • #9
fresh_42 said:
You should really calculate ## (aI+bN)^n ## by the binomial formula. With ##(...)^0=I## and ##N^2=0## you can almost write the answer without any calculation. Since ##[I,N]=0## you can also calculate ##e^{tA}=e^{atI}\cdot e^{tbN}##.
Yup I agree about the commutator part and splitting in 2 exponentials. Using only the [itex](Ia+Nb)^n[/itex] however, only helps you to calculate [itex]A^n[/itex].
[itex]e^{aI + bN} = e^{atI}\cdot e^{tbN} [/itex] (from [itex][N,I]=0[/itex] )
[itex]e^{atI}\cdot e^{tbN} = (\sum_n \frac{(at)^n}{n!} ) I^n \cdot (\sum_m \frac{(bt)^m N^m}{m!})[/itex]
[itex]e^{atI}\cdot e^{tbN} = e^{at} I \cdot (I + btN)= e^{at} I + bt e^{at} N[/itex]
 
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Related to Question regarding the exponential of a matrix

1. What is the exponential of a matrix?

The exponential of a matrix is a mathematical operation that is defined for square matrices. It is denoted by eA and is calculated by using the Taylor series expansion of the exponential function. The exponential of a matrix has many important applications in fields such as physics, engineering, and economics.

2. How is the exponential of a matrix calculated?

The exponential of a matrix is calculated using the Taylor series expansion of the exponential function. This involves summing an infinite series of terms, each of which involves powers of the matrix A. The calculation can be simplified for certain types of matrices, such as diagonal matrices or matrices with repeated eigenvalues.

3. What are the properties of the exponential of a matrix?

The exponential of a matrix has several important properties, including:

  • eA is always a square matrix of the same size as A.
  • If A and B are two matrices that commute (AB = BA), then eA+B = eAeB.
  • The exponential of a diagonal matrix is the matrix with the exponential of each diagonal element.
  • The exponential of a nilpotent matrix (a matrix whose nth power is the zero matrix) is the identity matrix plus the original matrix.

4. What are the applications of the exponential of a matrix?

The exponential of a matrix has many applications in various fields. Some examples include:

  • In physics, the exponential of a matrix is used to describe the evolution of a system over time, such as in quantum mechanics or fluid dynamics.
  • In engineering, the exponential of a matrix is used in control theory and signal processing.
  • In economics, the exponential of a matrix is used to model population growth and other dynamic systems.

5. Are there any real-world examples of the exponential of a matrix?

Yes, there are many real-world examples where the exponential of a matrix is used. One example is in the calculation of compound interest, where the exponential of a matrix is used to model the growth of an investment over time. Another example is in the calculation of the decay of radioactive materials, where the exponential of a matrix is used to model the rate of decay over time.

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