Proving Linear Dependence in Complex Matrix Space

In summary, the problem is to show that for a n by n matrix with complex coefficients, there exists an integer k that only depends on n, such that C, C^2, C^3, ..., C^k are linearly dependent. The proposed solution involves using i^4=i and Cayley-Hamilton's theorem. The asker also requests a link with an explanation, and suggests considering the possibility of extracting 7 linearly independent vectors from a vector space with dimension 6.
  • #1
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Homework Statement


If C is a n by n matrix with complex coefficient show that there is an integer k >0 that depends only on n such that C, C^2,C^3,...C^k are lin dependant.

The Attempt at a Solution



Its meant to be a full proof but the only idea I vaguely have is that i^4=i. I'm sure this is something standard to prove but I couldn't find it on google, so a link with an explanation would be appreciated.
 
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  • #2
I can't remember much linear algebra but I would try considering the following. Suppose the dimension of a vector space is 6. Is it possible to extract 7 linearly independent vectors from that vector space?
 
  • #3
Cayley-Hamilton tells us that C satisfies its own characteristic polynomial.
 

FAQ: Proving Linear Dependence in Complex Matrix Space

What is a complex matrix space?

A complex matrix space is a mathematical concept that refers to a collection of matrices that have complex numbers as their entries. This space can be represented as a set of matrices with the same dimensions, where the entries can be any complex number, including imaginary numbers.

What are the properties of a complex matrix space?

Some of the properties of a complex matrix space include closure, commutativity, associativity, and distributivity. This means that the set of matrices in this space can be added, multiplied, and scaled without leaving the space, and these operations follow the usual rules of algebra.

How is a complex matrix space different from a real matrix space?

A complex matrix space differs from a real matrix space in that the entries of the matrices in the former are complex numbers, while the entries in the latter are real numbers. This means that the operations of addition and multiplication in a complex matrix space involve complex arithmetic, while in a real matrix space, they only involve real arithmetic.

What are some applications of complex matrix space?

Complex matrix space has various applications in fields such as physics, engineering, and computer science. It is used in quantum mechanics to represent the states of quantum systems, in signal processing to analyze signals with complex values, and in computer graphics to transform and manipulate 3D objects.

How is a complex matrix space related to linear transformations?

A complex matrix space is closely related to linear transformations, as matrices can be used to represent these transformations. In particular, square matrices in a complex matrix space can represent linear transformations on vector spaces with complex entries. This allows for the use of matrix operations to analyze and manipulate linear transformations.

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