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Asked: 2024-09-27T05:01:42+05:30

How can students efficiently compute concise characteristic polynomials for 3×3 matrices in a coding competition?

I stumbled across this interesting challenge about characteristic polynomials, and I thought it would be fun to dive into it with a twist! Here’s the scenario:

Imagine you’re trying to help a group of budding mathematicians at a local college. They’ve been tasked with analyzing the characteristic polynomial of a 3×3 matrix, but as often happens in such scenarios, they’re getting a bit overwhelmed.

Here’s the thing: they need to compute the characteristic polynomial, but they want to do it in the most efficient way possible using code. However, rather than doing it the traditional way, they’re hoping to make it a bit more engaging by turning it into a competitive challenge.

So, what if I threw in some extra flair? Let’s say this group loves puzzles, and to make it a bit more engaging, I want to challenge them to write a function that not only computes the characteristic polynomial but also does so while minimizing the character count of their code. The catch? They need to work with various templates of the 3×3 matrix that could represent different types of transformations, like rotations, scalings, and reflections.

The key components of their task would include input matrices, the mathematical logic to derive the characteristic polynomial, and of course, ensuring their code is as succinct as possible.

As a kick-off, just to get the creative juices flowing, they could start with a simple matrix like:
\[
\begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\]

Then, sportingly, I want them to see who can come up with the shortest code that calculates the characteristic polynomial. Maybe they could share their language of choice, compare outputs, and possibly even explain any nifty tricks they used to save characters.

So, what do you think? Could you help me brainstorm some fun ideas or tips for these students? Maybe ways they can approach the problem or even classical examples of characteristic polynomials they could explore? I’m excited to see what you all come up with!

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    2 Answers

    1. To tackle the challenge of computing the characteristic polynomial of a 3×3 matrix in a succinct manner, the group can start by using Python, which offers various libraries to simplify matrix operations. A concise approach involves utilizing NumPy, which allows them to leverage its built-in functions. Here’s a straightforward example of how they can define a function that computes the characteristic polynomial:

      import numpy as np
def char_poly(matrix):
    return np.poly(np.linalg.eigvals(matrix))

      This function utilizes NumPy’s `eigvals` to find the eigenvalues of the input matrix, which are essential for determining the characteristic polynomial. The `poly` function then constructs the polynomial coefficients from these eigenvalues. This code is efficient and demonstrates how they can minimize character count while still achieving the desired output. To explore further, they might consider matrices representing different transformations, such as:

      r = np.array([[0,-1,0],[1,0,0],[0,0,1]]) # Rotation
s = np.array([[2,0,0],[0,3,0],[0,0,1]]) # Scaling
f = np.array([[1,0,0],[0,-1,0],[0,0,1]]) # Reflection

      By comparing the characteristic polynomials of these transformation matrices, they can gain insights into how different properties affect the final output, encouraging a deeper understanding of the subject.

    2. Fun Challenge for Characteristic Polynomials!

      Hey team! Here’s a fun idea to tackle the characteristic polynomial of a 3×3 matrix. Let’s keep it simple and exciting!

      Step 1: Input Your Matrix

      Start with an example matrix:

          [[1, 2, 3],
     [0, 1, 4],
     [5, 6, 0]]

      Step 2: Calculate the Characteristic Polynomial

      Here’s a super compact Python function to calculate the characteristic polynomial:

          import numpy as np
    def char_poly(A):
        return np.poly(np.linalg.eigvals(A))

      Step 3: Make a Fun Code Competition!

      Challenge your friends to write their version. Who can do it in fewer characters? Here are some things to encourage:

      • Try Different Languages: Python, JavaScript, C++, etc.
      • Concepts to Explore: Matrix transformations like rotations, scalings, reflections.
      • Share Tricks: Use lambda functions, list comprehensions, or anything to squeeze those characters!

      Step 4: Classical Examples

      It’s cool to look at classic matrices and their polynomials:

          Rotation 90°: [[0, -1, 0], 
                  [1,  0, 0], 
                  [0,  0, 1]]
    
    Scaling: [[2, 0, 0], 
              [0, 2, 0], 
              [0, 0, 1]]
    
    Reflection: [[1, 0, 0], 
                 [0, -1, 0], 
                 [0, 0, 1]]

      Good luck, have fun, and may the shortest code win!

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