Show That: A) The Only Matrix Similar To I N I_n I N ​ Is I N I_n I N ​ Itself. B) The Only Matrix Similar To The Zero Matrix Is The Zero Matrix Itself.2. A) Define A Diagonal Matrix. B) Show That The Product Of Two Diagonal

Introduction

In linear algebra, two matrices are said to be similar if one can be transformed into the other through a series of elementary row operations. This concept is crucial in understanding the properties and behavior of matrices. In this article, we will explore the similarity of matrices, specifically focusing on the properties of the identity matrix and the zero matrix. We will also delve into the definition and properties of diagonal matrices.

Similarity of Matrices

a) The only matrix similar to $I_n$ is $I_n$ itself.

To show that the only matrix similar to $I_n$ is $I_n$ itself, we need to understand the concept of similarity between matrices. Two matrices $A$ and $B$ are said to be similar if there exists an invertible matrix $P$ such that $A = PBP^{-1}$. In other words, $A$ and $B$ are similar if they can be transformed into each other through a series of elementary row operations.

Now, let's consider the identity matrix $I_n$. We want to show that the only matrix similar to $I_n$ is $I_n$ itself. Suppose that $A$ is a matrix similar to $I_n$, i.e., $A = P I_n P^{-1}$ for some invertible matrix $P$. We can rewrite this equation as $A = P I_n P^{-1} = P I_n (P^{-1} P) = P I_n I_n = P I_n$. Since $P$ is invertible, we can multiply both sides of the equation by $P^{-1}$ to get $P^{-1} A P = I_n$. This shows that $A$ is similar to $I_n$.

However, we can also show that $A$ must be equal to $I_n$. Since $A$ is similar to $I_n$, we know that $A$ and $I_n$ have the same characteristic polynomial. The characteristic polynomial of $I_n$ is $x^n - 1$, which has $n$ distinct roots. Since $A$ and $I_n$ have the same characteristic polynomial, $A$ must also have $n$ distinct roots. However, the only matrix with $n$ distinct roots is the identity matrix $I_n$ itself. Therefore, we conclude that the only matrix similar to $I_n$ is $I_n$ itself.

b) The only matrix similar to the zero matrix is the zero matrix itself.

To show that the only matrix similar to the zero matrix is the zero matrix itself, we can use a similar argument as above. Suppose that $A$ is a matrix similar to the zero matrix, i.e., $A = P 0_n P^{-1}$ for some invertible matrix $P$. We can rewrite this equation as $A = P 0_n P^{-1} = P 0_n (P^{-1} P) = P 0_n I_n = P 0_n$. Since $P$ is invertible, we can multiply both sides of the equation by $P^{-1}$ to get $P^{-1} A P = 0_n$. This shows that $A$ is similar to the zero matrix.

However, we can also show that $A$ must be equal to the zero matrix. Since $A$ is similar to the zero matrix, we know that $A$ and the zero matrix have the same characteristic polynomial. The characteristic polynomial of the zero matrix is $x^n$, which has only one root, namely $0$. Since $A$ and the zero matrix have the same characteristic polynomial, $A$ must also have only one root, namely $0$. However, the only matrix with only one root is the zero matrix itself. Therefore, we conclude that the only matrix similar to the zero matrix is the zero matrix itself.

Diagonal Matrices

a) Define a diagonal matrix.

A diagonal matrix is a square matrix whose entries are zero except for the entries on the main diagonal. In other words, a diagonal matrix is a matrix of the form:

$$D = \begin{bmatrix} d_{11} & 0 & \cdots & 0 \\ 0 & d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} \end{bmatrix}$$

where $d_{ii}$ are the entries on the main diagonal.

b) Show that the product of two diagonal matrices is a diagonal matrix.

To show that the product of two diagonal matrices is a diagonal matrix, we can use the following argument. Suppose that $D_1$ and $D_2$ are two diagonal matrices, i.e.,

$$D_1 = \begin{bmatrix} d_{11} & 0 & \cdots & 0 \\ 0 & d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} \end{bmatrix}$$

and

$$D_2 = \begin{bmatrix} e_{11} & 0 & \cdots & 0 \\ 0 & e_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & e_{nn} \end{bmatrix}$$

We can compute the product $D_1 D_2$ as follows:

$$D_1 D_2 = \begin{bmatrix} d_{11} e_{11} & 0 & \cdots & 0 \\ 0 & d_{22} e_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} e_{nn} \end{bmatrix}$$

This shows that the product of two diagonal matrices is a diagonal matrix.

Conclusion

Q: What is the definition of similarity between matrices?

A: Two matrices $A$ and $B$ are said to be similar if there exists an invertible matrix $P$ such that $A = PBP^{-1}$. In other words, $A$ and $B$ are similar if they can be transformed into each other through a series of elementary row operations.

Q: How do you show that the only matrix similar to $I_n$ is $I_n$ itself?

A: To show that the only matrix similar to $I_n$ is $I_n$ itself, we need to understand that if $A$ is similar to $I_n$, then $A$ and $I_n$ have the same characteristic polynomial. The characteristic polynomial of $I_n$ is $x^n - 1$, which has $n$ distinct roots. Since $A$ and $I_n$ have the same characteristic polynomial, $A$ must also have $n$ distinct roots. However, the only matrix with $n$ distinct roots is the identity matrix $I_n$ itself.

Q: How do you show that the only matrix similar to the zero matrix is the zero matrix itself?

A: To show that the only matrix similar to the zero matrix is the zero matrix itself, we can use a similar argument as above. If $A$ is similar to the zero matrix, then $A$ and the zero matrix have the same characteristic polynomial. The characteristic polynomial of the zero matrix is $x^n$, which has only one root, namely $0$. Since $A$ and the zero matrix have the same characteristic polynomial, $A$ must also have only one root, namely $0$. However, the only matrix with only one root is the zero matrix itself.

Q: What is a diagonal matrix?

A: A diagonal matrix is a square matrix whose entries are zero except for the entries on the main diagonal. In other words, a diagonal matrix is a matrix of the form:

$$D = \begin{bmatrix} d_{11} & 0 & \cdots & 0 \\ 0 & d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} \end{bmatrix}$$

where $d_{ii}$ are the entries on the main diagonal.

Q: How do you show that the product of two diagonal matrices is a diagonal matrix?

A: To show that the product of two diagonal matrices is a diagonal matrix, we can use the following argument. Suppose that $D_1$ and $D_2$ are two diagonal matrices, i.e.,

$$D_1 = \begin{bmatrix} d_{11} & 0 & \cdots & 0 \\ 0 & d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} \end{bmatrix}$$

and

$$D_2 = \begin{bmatrix} e_{11} & 0 & \cdots & 0 \\ 0 & e_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & e_{nn} \end{bmatrix}$$

We can compute the product $D_1 D_2$ as follows:

$$D_1 D_2 = \begin{bmatrix} d_{11} e_{11} & 0 & \cdots & 0 \\ 0 & d_{22} e_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_{nn} e_{nn} \end{bmatrix}$$

This shows that the product of two diagonal matrices is a diagonal matrix.

Q: What are some applications of similarity of matrices and diagonal matrices?

A: Similarity of matrices and diagonal matrices have numerous applications in linear algebra and beyond. Some examples include:

• Eigenvalues and eigenvectors: Similarity of matrices is used to find eigenvalues and eigenvectors of a matrix.
• Diagonalization: Diagonal matrices are used to diagonalize a matrix, which is a process of transforming a matrix into a diagonal matrix.
• Linear transformations: Similarity of matrices is used to study linear transformations, which are functions that preserve the operations of vector addition and scalar multiplication.
• Graph theory: Diagonal matrices are used to study graph theory, which is the study of graphs and their properties.

Conclusion

In this Q&A article, we have explored the similarity of matrices and diagonal matrices, including the definition of similarity, the properties of diagonal matrices, and some applications of these concepts. We hope that this article has provided a helpful overview of these important topics in linear algebra.