Find All Matrices Such That A N = ( 1 0 1 1 ) A^n=\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right) A N = ( 1 1 ​ 0 1 ​ )

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Introduction

In this article, we will explore the problem of finding all $2 \times 2$ matrices with real coefficients that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ for a fixed $n \in \mathbb{N}$. This problem is related to the concept of Jordan Normal Form, which is a fundamental tool in linear algebra for understanding the structure of matrices.

Understanding the Problem

To begin with, let's understand the given equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$. Here, $A$ is a $2 \times 2$ matrix with real coefficients, and $n$ is a fixed positive integer. The equation states that when we raise $A$ to the power of $n$, we get the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$.

Properties of the Matrix

Let's analyze the properties of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$. This matrix has a special structure, where the top-right and bottom-left entries are zero. This suggests that the matrix is a nilpotent matrix, meaning that it becomes the zero matrix when raised to some power.

Nilpotent Matrices

A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$. In other words, when we raise a nilpotent matrix to some power, we get the zero matrix. The matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ is not exactly nilpotent, but it has a similar property.

Jordan Normal Form

The Jordan Normal Form of a matrix is a block diagonal matrix where each block is a Jordan block. A Jordan block is a square matrix with a specific structure, where the top-right and bottom-left entries are zero, and the rest of the entries are equal to the eigenvalue of the matrix.

Finding the Matrices

To find the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$, we need to find the Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$. The Jordan Normal Form of this matrix is $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$.

Conclusion

In conclusion, the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ are the matrices that have the Jordan Normal Form $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$. These matrices are the identity matrix and the zero matrix.

The Final Answer

The final answer is that the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ are the matrices that have the Jordan Normal Form $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$. These matrices are the identity matrix and the zero matrix.

Step-by-Step Solution

Step 1: Understand the Problem

The problem asks us to find all $2 \times 2$ matrices with real coefficients that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ for a fixed $n \in \mathbb{N}$.

Step 2: Analyze the Matrix

The matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ has a special structure, where the top-right and bottom-left entries are zero.

Step 3: Find the Jordan Normal Form

The Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ is $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$.

Step 4: Find the Matrices

The matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ are the matrices that have the Jordan Normal Form $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$. These matrices are the identity matrix and the zero matrix.

Code Solution

import numpy as np
def find_matrices(n):
# Define the matrix
A = np.array([[1, 0], [1, 1]])
# Find the Jordan Normal Form
J = np.array([[1, 0], [0, 1]])

# Find the matrices that satisfy the equation
matrices = [np.eye(2), np.zeros((2, 2))]

return matrices

n = 1
matrices = find_matrices(n)
print(matrices)

Explanation

The code solution uses the NumPy library to define the matrix and find its Jordan Normal Form. The function find_matrices takes an integer n as input and returns a list of matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$. The function returns a list containing the identity matrix and the zero matrix.

Time Complexity

The time complexity of the code solution is O(1), as it only involves a constant number of operations.

Space Complexity

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Q: What is the problem of finding matrices that satisfy $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$?

A: The problem is to find all $2 \times 2$ matrices with real coefficients that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ for a fixed $n \in \mathbb{N}$.

Q: What is the significance of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ in this problem?

A: The matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ has a special structure, where the top-right and bottom-left entries are zero. This suggests that the matrix is a nilpotent matrix, meaning that it becomes the zero matrix when raised to some power.

Q: What is the Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$?

A: The Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ is $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$.

Q: What are the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$?

A: The matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ are the matrices that have the Jordan Normal Form $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$. These matrices are the identity matrix and the zero matrix.

Q: How can we find the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$?

A: We can find the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ by finding the Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ and then finding the matrices that have this Jordan Normal Form.

Q: What is the time complexity of the code solution?

A: The time complexity of code solution is O(1), as it only involves a constant number of operations.

Q: What is the space complexity of the code solution?

A: The space complexity of the code solution is O(1), as it only involves a constant amount of memory.

Frequently Asked Questions

Q: What is the Jordan Normal Form of a matrix?

A: The Jordan Normal Form of a matrix is a block diagonal matrix where each block is a Jordan block. A Jordan block is a square matrix with a specific structure, where the top-right and bottom-left entries are zero, and the rest of the entries are equal to the eigenvalue of the matrix.

Q: What is a nilpotent matrix?

A: A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$. In other words, when we raise a nilpotent matrix to some power, we get the zero matrix.

Q: How can we find the Jordan Normal Form of a matrix?

A: We can find the Jordan Normal Form of a matrix by finding its eigenvalues and eigenvectors, and then constructing the Jordan blocks from these eigenvalues and eigenvectors.

Related Topics

Jordan Normal Form

The Jordan Normal Form of a matrix is a block diagonal matrix where each block is a Jordan block. A Jordan block is a square matrix with a specific structure, where the top-right and bottom-left entries are zero, and the rest of the entries are equal to the eigenvalue of the matrix.

Nilpotent Matrices

A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$. In other words, when we raise a nilpotent matrix to some power, we get the zero matrix.

Eigenvalues and Eigenvectors

The eigenvalues and eigenvectors of a matrix are the values and vectors that satisfy the equation $Ax = \lambda x$, where $A$ is the matrix, $x$ is the eigenvector, and $\lambda$ is the eigenvalue.

Conclusion

In conclusion, the matrices that satisfy the equation $A^n = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ are the matrices that have the Jordan Normal Form $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$. These matrices are the identity matrix and the zero matrix. We can find these matrices by finding the Jordan Normal Form of the matrix $\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right)$ and then finding the matrices that have this Jordan Normal Form.