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 Diagonal Matrix. B) Show That The Product Of Two Diagonal Matrices Is A

Introduction

In linear algebra, two matrices are said to be similar if one can be transformed into the other through a change of basis. This concept is crucial in understanding the properties of matrices and their behavior under different transformations. 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 change of basis.

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 $A = P I_n P^{-1} = P I_n P^{-1} P P^{-1} = P I_n P^{-1} P = P I_n P$.

Now, let's consider the product $P I_n P$. Since $P$ is invertible, we can multiply both sides of the equation by $P^{-1}$ to get $P^{-1} P I_n P = I_n P$. Simplifying this equation, we get $I_n P = P$. This implies that $P$ is equal to the identity matrix $I_n$.

Therefore, we have shown that if $A$ is similar to $I_n$, then $A$ must be equal to $I_n$ itself. This means 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 follow 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 $A = P 0_n P^{-1} = P 0_n P^{-1} P P^{-1} = P 0_n P^{-1} P = P 0_n P$.

Now, let's consider the product $P 0_n P$. Since $P$ is invertible, we can multiply both sides of the equation by $P^{-1}$ to get $P^{-1} P 0_n P = 0_n P$. Simplifying this equation, we get $0_n P = P$. This implies that $P$ is equal to the zero matrix $0_n$.

Therefore, we have shown that if $A$ is similar to the zero matrix, then $A$ must be equal to the zero matrix itself. This means that the only matrix similar to the zero matrix is the zero matrix itself.

Diagonal Matrices

a) Define diagonal matrix

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero. 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 consider two diagonal matrices $D_1$ and $D_2$ of the form:

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

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

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

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

As we can see, the product $D_1 D_2$ is also a diagonal matrix. This means that the product of two diagonal matrices is a diagonal matrix.

Conclusion

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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 change of basis.

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 the concept of similarity between matrices. 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 $A = P I_n P^{-1} = P I_n P^{-1} P P^{-1} = P I_n P^{-1} P = P I_n P$. This implies that $P$ is equal to the identity matrix $I_n$.

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 follow 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 $A = P 0_n P^{-1} = P 0_n P^{-1} P P^{-1} = P 0_n P^{-1} P = P 0_n P$. This implies that $P$ is equal to the zero matrix $0_n$.

Q: What is a diagonal matrix?

A: A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero. 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 consider two diagonal matrices $D_1$ and $D_2$ of the form:

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

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

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

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

As we can see, the product $D_1 D_2$ is also a diagonal matrix.

Q: What are some common properties of diagonal matrices?

A: Some common properties of diagonal matrices include:

• The product of two diagonal matrices is a diagonal matrix.
• The inverse of a diagonal matrix is a diagonal matrix.
• The determinant of a diagonal matrix is the product of the entries on the main diagonal.

Q: How do you find the inverse of a diagonal matrix?

A: To find the inverse of a diagonal matrix, we can simply take the reciprocal of each entry on the main diagonal. For example, if we have a diagonal matrix:

$$D = \begin{bmatrix} 2 & 0 & \cdots & 0 \\ 0 & 3 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 5 \end{bmatrix}$$

then the inverse of $D$ is:

$$D^{-1} = \begin{bmatrix} \frac{1}{2} & 0 & \cdots & 0 \\ 0 & \frac{1}{3} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \frac{1}{5} \end{bmatrix}$$

Q: What are some common applications of diagonal matrices?

A: Some common applications of diagonal matrices include:

• Finding the inverse of a matrix.
• Computing the determinant of a matrix.
• Solving systems of linear equations.

Conclusion

In this Q&A article, we have explored the concept of similarity between matrices, specifically focusing on the properties of the identity matrix and the zero matrix. We have also defined diagonal matrices and shown that the product of two diagonal matrices is a diagonal matrix. These results provide valuable insights into the properties of matrices and their behavior under different transformations.