Log of 2x2 Matrix: Solving e^A= Ʃ (A^k)/k!

In summary, the conversation was about finding the logarithm of a 2x2 matrix and the use of e^X = A and X = log A. The suggested method was to use matrix diagonalization, as described in the Wikipedia article.
  • #1
jhendren
34
0

Homework Statement



log of 2x2 matrix [4 3]
[3 4]

Homework Equations



e^X= A X= log A

The Attempt at a Solution



e^A= Ʃ (A^k)/k!

then what to do?
 
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  • #2
jhendren said:

Homework Statement



log of 2x2 matrix [4 3]
[3 4]

Homework Equations



e^X= A X= log A

The Attempt at a Solution



e^A= Ʃ (A^k)/k!

then what to do?

I'm not sure that what you've done will be much help. Have you learned matrix diagonalization yet? If so, your matrix is diagonalizable. See http://en.wikipedia.org/wiki/Logarithm_of_a_matrix, especially the section titled "Calculating the logarithm of a diagonalizable matrix".
 

FAQ: Log of 2x2 Matrix: Solving e^A= Ʃ (A^k)/k!

What is a log of a 2x2 matrix?

A log of a 2x2 matrix is a mathematical function that represents the logarithm of a 2x2 matrix. It is often used in the study of linear algebra and differential equations.

How do you solve e^A= Ʃ (A^k)/k! for a 2x2 matrix?

To solve this equation, you can use the Taylor series expansion of e^A, which states that e^A = Ʃ (A^k)/k!. By equating this to the given equation, you can then solve for A using matrix operations.

What is the significance of solving e^A= Ʃ (A^k)/k! for a 2x2 matrix?

Solving this equation allows you to find the logarithm of a 2x2 matrix, which is useful in many applications, such as in solving differential equations and in calculating the matrix exponential.

Can this equation be solved for matrices of other sizes?

Yes, this equation can be solved for matrices of any size. However, the process becomes more complex as the size of the matrix increases.

How is the solution to this equation used in scientific research?

The solution to this equation is used in a variety of scientific fields, including physics, engineering, and economics. It can help in solving differential equations, predicting future values of a system, and understanding the behavior of complex systems.

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