Exploring Relationship b/w C.D^n.F and A^n

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In summary, the conversation discusses the relationship between matrices C, D, F, and A, where F = C^-1 and C.D.F = A. The discussion explores different ways to prove that C.D^n.F = A^n, but concludes that without any additional properties, the relationship can only be simplified to show that A and D are similar matrices with the same eigenvalues.
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Homework Statement


Given 3 matrices C, D, F and another matrix A, can i say anything in general about the relationship between C.D^n.F and A^n if i know that F = C^-1 and that C.D.F = A.


Homework Equations





The Attempt at a Solution


For example,
If C.D.F = A then (C.D.F)^2 = A^2 and then C.D.F.C.D.F = A^2. Since F = C^-1 i can rewrite as C.D.D.F = A^2 and so C.D^2.F = A^2. I could use induction on n to show that C.D^n.F = A^n. The thing is i don't see what this says about the general relationship. It would just prove equality. Another way would be something like, C.D.F = A so F.C.D.F.C = F.A.C then, D = F.A.C and substituting that into C.D.F would give, C.D.F = C.F.A.C.F. Again this could be used to prove that C.D^n.F = A^n for all n. And again I'm not really interested in proving that they are equal. Is there some rule of matrix multiplication, that I'm not aware of, that could be used to describe the relationship? I'm a little lost here...
 
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  • #2
Without any extra special properties, there's no further simplification of the relationship. However, since A = C D C-1, we know that A and D are similar matrices. That is, they refer to the same linear transformation, but in different bases. C-1 is the matrix that determines the change of basis. In line with this interpretation, it's also simple to show that A and D have the same eigenvalues.
 
  • #3
Oh, thanks a lot for the reply. I've never heard of similar matrices before (this is my first week of linear algebra), but i looked it up and it got me going. So yeah, thanks.
 

1. What is the relationship between C.D^n.F and A^n?

The relationship between C.D^n.F and A^n is a mathematical one, where C.D^n.F represents a set of complex numbers raised to the n-th power, and A^n represents a set of real numbers raised to the n-th power. In simpler terms, both sets involve taking a number and multiplying it by itself n times, but C.D^n.F involves using complex numbers while A^n involves using real numbers.

2. How is the relationship between C.D^n.F and A^n explored?

The relationship between C.D^n.F and A^n is explored through mathematical analysis and computations. This involves studying the properties and behaviors of complex and real numbers when raised to different powers, and looking for patterns and connections between the two sets. It also involves using mathematical equations and formulas to represent and manipulate these relationships.

3. What are some practical applications of exploring the relationship between C.D^n.F and A^n?

Understanding the relationship between C.D^n.F and A^n has various practical applications in fields such as physics, engineering, and computer science. For example, in signal processing, complex numbers are often used to represent electromagnetic waves, and analyzing the relationship between C.D^n.F and A^n can help in understanding and manipulating these waves. In computer graphics, the use of complex numbers in representing transformations can also benefit from exploring this relationship.

4. Is there a general formula for the relationship between C.D^n.F and A^n?

Yes, there is a general formula for the relationship between C.D^n.F and A^n, known as the De Moivre's formula. It states that (C.D)^n.F = (A^n)cos(nF) + i(A^n)sin(nF), where i is the imaginary unit and n is any positive integer. This formula allows for the conversion between complex and real numbers raised to different powers and is a fundamental tool in exploring the relationship between the two sets.

5. What are some potential challenges in exploring the relationship between C.D^n.F and A^n?

One potential challenge in exploring the relationship between C.D^n.F and A^n is the complexity of the subject matter. Complex numbers can be difficult to understand and manipulate, and their relationship with real numbers can be even more complex. Additionally, different operations involving these numbers can lead to a wide range of results, making it challenging to establish a clear and definitive relationship. Another challenge is the potential for errors in calculations, which can affect the accuracy of the results and conclusions.

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