E^A matrix power series (eigen values, diagonalizable)

In summary, the conversation discusses finding an expression for e^A using eigenvalues and eigenvectors, and the attempt at solving it by multiplying out the given power series. However, the speaker did not receive credit for this solution and is seeking help in finding the eigenvectors.
  • #1
Fellowroot
92
0

Homework Statement


Find an expression for e^A with the powerseries shown in the image linked

Homework Equations


I know you have to use eigen values and eigen vectors and a diagonal matrix

The Attempt at a Solution


What I did was just try to actually multiply out the infinite series given. I took it out to about 3 terms and said on my quiz that the rest will eventually go to zero so that this series will converge. However I got zero credit for this solution.

I know how to get eigen values, but I just need help finding out how to get Q.

Thanks.
 

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  • #2
Can you find the eigenvectors? Those will typically be the columns of ##Q##.
 

FAQ: E^A matrix power series (eigen values, diagonalizable)

1. What is an E^A matrix power series?

An E^A matrix power series is a mathematical representation of a function that raises a square matrix E to the power of a real number A. It is written as E^A = I + A*E + (A^2/2!)*E^2 + (A^3/3!)*E^3 + ... , where I represents the identity matrix.

2. What are eigen values and how are they related to E^A matrix power series?

Eigen values are the set of numbers that represent the scale by which a matrix E stretches a vector in a particular direction. They are related to E^A matrix power series because they are used to calculate the power series, and can also help determine if the matrix is diagonalizable.

3. Can an E^A matrix power series have complex eigen values?

Yes, an E^A matrix power series can have complex eigen values. This is because the values of A can be complex numbers, and the power series can still be calculated using the same formula as for real numbers.

4. What does it mean for a matrix to be diagonalizable?

A matrix is considered diagonalizable if it can be transformed into a diagonal matrix through similarity transformations. This means that the matrix can be represented by a simpler form, making it easier to perform calculations.

5. How can I determine if a matrix is diagonalizable?

A matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the dimension of the matrix. This means that the number of unique eigen values must be equal to the dimension of the matrix. Additionally, the matrix must be non-defective, meaning there are enough eigenvectors to span the entire space.

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