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 delve into the problem of finding all $2 \times 2$ matrices with real coefficients that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ for a fixed $n \in \mathbb{N}$. This problem is related to the concept of Jordan Normal Form, which is a fundamental concept in linear algebra.

Understanding the Problem

To begin with, let's understand the problem at hand. We are given a $2 \times 2$ matrix $A$ with real coefficients, and we need to find all such matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ for a fixed $n \in \mathbb{N}$. This means that when we raise the matrix $A$ to the power of $n$, we get the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.

Jordan Normal Form

The Jordan Normal Form is a fundamental concept in linear algebra that helps us understand the structure of matrices. A matrix is said to be in Jordan Normal Form if it can be transformed into a block diagonal matrix, where each block is a Jordan block. A Jordan block is a square matrix with a single eigenvalue on the main diagonal, and ones on the superdiagonal.

Eigenvalues and Eigenvectors

To find the matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$, we need to find the eigenvalues and eigenvectors of the matrix $A$. The eigenvalues of a matrix are the values that the matrix can take when it is multiplied by a non-zero vector. The eigenvectors of a matrix are the non-zero vectors that, when multiplied by the matrix, result in a scaled version of the same vector.

Finding the Eigenvalues

To find the eigenvalues of the matrix $A$, we need to solve the characteristic equation $|A - \lambda I| = 0$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. For a $2 \times 2$ matrix, the characteristic equation is given by:

$$\begin{vmatrix} a - \lambda & b \\ c & d - \lambda \end{vmatrix} = 0$$

Expanding the determinant, we get:

$$(a - \lambda)(d - \lambda) - bc = 0$$

Simplifying the equation, we get:

$$\lambda^2 - (a + d)\lambda + (ad - bc) = 0$$

Finding the Eigenvectors

Once we have found the eigenvalues, we need to find the corresponding eigenvectors. The eigenvectors are the non-zero vectors that, when multiplied by the matrix, result in a scaled version of the same vector. To find the eigenvectors, we need to solve the equation $(A - \lambda I)v = 0$, where $v$ is the eigenvector.

Solving the Equation

To solve the equation $(A - \lambda I)v = 0$, we need to find the values of $v$ that satisfy the equation. Let's assume that the eigenvalue is $\lambda = 1$. Then, the equation becomes:

$$(A - I)v = 0$$

Simplifying the equation, we get:

$$\begin{pmatrix} a - 1 & b \\ c & d - 1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

Finding the Solutions

To find the solutions to the equation, we need to find the values of $v_1$ and $v_2$ that satisfy the equation. Let's assume that $v_1 = 1$. Then, we get:

$$(a - 1)v_1 + bv_2 = 0$$

Simplifying the equation, we get:

$$(a - 1) + bv_2 = 0$$

Solving for $v_2$, we get:

$$v_2 = \frac{a - 1}{b}$$

Finding the Matrices

Now that we have found the eigenvalues and eigenvectors, we can find the matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$. Let's assume that the eigenvalue is $\lambda = 1$. Then, the matrix $A$ can be written as:

$$A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$

Conclusion

In this article, we have found the matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$. We have used the concept of Jordan Normal Form and eigenvalues and eigenvectors to find the solutions. The matrices that satisfy the condition are given by:

$$A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$

References

• [1] Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis. Cambridge University Press.
• [2] Gantmacher, F. R. (1959). The Theory of Matrices. Chelsea Publishing Company.
• [3] Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice-Hall.

Additional Information

• The problem of finding the matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ is related to the concept of Jordan Normal Form.
• The eigenvalues and eigenvectors of a matrix are used to find the solutions to the equation $(A - \lambda I)v = 0$.
• The matrices that satisfy the condition are given by $A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$.

Final Answer

The final answer is $\boxed{\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}}$.

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Q: What is the Jordan Normal Form, and how is it related to the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: The Jordan Normal Form is a fundamental concept in linear algebra that helps us understand the structure of matrices. A matrix is said to be in Jordan Normal Form if it can be transformed into a block diagonal matrix, where each block is a Jordan block. A Jordan block is a square matrix with a single eigenvalue on the main diagonal, and ones on the superdiagonal. The Jordan Normal Form is related to the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ because the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ is in Jordan Normal Form.

Q: How do we find the eigenvalues and eigenvectors of a matrix?

A: To find the eigenvalues and eigenvectors of a matrix, we need to solve the characteristic equation $|A - \lambda I| = 0$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. For a $2 \times 2$ matrix, the characteristic equation is given by:

$$(a - \lambda)(d - \lambda) - bc = 0$$

Simplifying the equation, we get:

$$\lambda^2 - (a + d)\lambda + (ad - bc) = 0$$

Q: What is the relationship between the eigenvalues and the eigenvectors of a matrix?

A: The eigenvalues and eigenvectors of a matrix are related in the following way: the eigenvalues are the values that the matrix can take when it is multiplied by a non-zero vector, and the eigenvectors are the non-zero vectors that, when multiplied by the matrix, result in a scaled version of the same vector.

Q: How do we find the solutions to the equation $(A - \lambda I)v = 0$?

A: To find the solutions to the equation $(A - \lambda I)v = 0$, we need to find the values of $v$ that satisfy the equation. Let's assume that the eigenvalue is $\lambda = 1$. Then, the equation becomes:

$$(A - I)v = 0$$

Simplifying the equation, we get:

$$\begin{pmatrix} a - 1 & b \\ c & d - 1 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

Q: What is the relationship between the matrix $A$ and the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: The matrix $A$ is related to the matrix $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ in the following way: the $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ is a Jordan block, and the matrix $A$ can be written as:

$$A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$

Q: What is the final answer to the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: The final answer to the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ is:

$$A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$$

Q: What are some additional resources that can be used to learn more about the Jordan Normal Form and the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: Some additional resources that can be used to learn more about the Jordan Normal Form and the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ include:

• [1] Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis. Cambridge University Press.
• [2] Gantmacher, F. R. (1959). The Theory of Matrices. Chelsea Publishing Company.
• [3] Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice-Hall.

Q: What is the significance of the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: The problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ is significant because it helps us understand the structure of matrices and the relationship between the eigenvalues and eigenvectors of a matrix. It also provides a way to find the solutions to the equation $(A - \lambda I)v = 0$.

Q: Can the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ be generalized to higher-dimensional matrices?

A: Yes, the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ can be generalized to higher-dimensional matrices. However, the solution to the problem will be more complex and will require the use of more advanced mathematical techniques.

Q: What are some potential applications of the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: Some potential applications of the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & \\ 1 & 1 \end{pmatrix}$ include:

• Cryptography: The problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ can be used to develop new cryptographic algorithms and protocols.
• Signal Processing: The problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ can be used to develop new signal processing algorithms and techniques.
• Machine Learning: The problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ can be used to develop new machine learning algorithms and techniques.

Q: What are some potential future research directions for the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$?

A: Some potential future research directions for the problem of finding matrices that satisfy the condition $A^n = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ include:

• Generalizing the problem to higher-dimensional matrices: Developing new mathematical techniques and algorithms to solve the problem for higher-dimensional matrices.
• Developing new applications for the problem: Developing new applications for the problem in fields such as cryptography, signal processing, and machine learning.
• Investigating the relationship between the problem and other mathematical concepts: Investigating the relationship between the problem and other mathematical concepts such as group theory and representation theory.