Solve Idempotent Matrix Inequality: n≥p-1 | Artin's Algebra

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In summary, the problem states that for a prime number p and an nxn integer matrix A, where A is not the identity and A^p = I, we need to prove that n is greater than or equal to p-1. This is in the factorization chapter, specifically the section on Explicit Factorization of Polynomials. A possible approach is to show that A is invertible and use the fact that A^(p+1) - A = 0.
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Dunkle
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Homework Statement



This is from Chapter 11 of Artin's Algebra:

Let p be a prime, and let A (not the identity) be an nxn integer matrix such that [tex]A^{p}=I[/tex]. Prove that [tex]n \geq p-1.[/tex]

Homework Equations



This is in the factorization chapter, and the section is called Explicit Factorization of Polynomials.

The Attempt at a Solution



I don't even know where to begin. I'm guessing I need to somehow get a polynomial equation, but like I said, I don't really know where to start. Any help would be greatly appreciated!
 
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haven't got there yet, but hopefully these help you get started

clearly A is invertible as A^(p-1) = A^-1

also note that
A^(p+1) - A = A(A^p - I ) = 0
 

1. What is an idempotent matrix?

An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. In other words, the matrix remains unchanged after multiple multiplications with itself.

2. How do you determine if a matrix is idempotent?

A matrix is idempotent if it satisfies the condition A2 = A, where A is the matrix itself. In other words, the matrix must be equal to its own square.

3. Why is the inequality n≥p-1 used in solving idempotent matrix inequalities?

This inequality is used to ensure that there are enough linearly independent rows in the matrix to satisfy the idempotent condition. If n=p-1, there are not enough rows to form an idempotent matrix.

4. Can you provide an example of solving an idempotent matrix inequality using Artin's Algebra?

Yes, for example, if we have the matrix A = [1 0; 0 0] and we want to find the values of n and p that satisfy the idempotent condition, we can use Artin's Algebra to solve the inequality n≥p-1. In this case, n=2 and p=2 would satisfy the condition.

5. What are the applications of idempotent matrices in real life?

Idempotent matrices have various applications in fields such as statistics, engineering, and computer science. They are used in linear regression, Markov chains, and image processing, among others. In statistics, idempotent matrices are used to represent projections, which are used to remove noise and focus on the important features of a dataset. In engineering and computer science, idempotent matrices are used in control systems and algorithms that require repetitive operations.

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