# Question about hollow matrix and diagonalization

• pizzamakeren
In summary, a hollow matrix is a square matrix with zeros along the main diagonal and non-zero elements elsewhere. It differs from a regular matrix in that it has fewer non-zero elements and a specific structure that can be useful for certain calculations. Diagonalization of a matrix is the process of finding a diagonal matrix that is similar to the original matrix, and a hollow matrix can also be diagonalized. Diagonalization is important because it simplifies the matrix and reveals important information about its eigenvalues and eigenvectors.
pizzamakeren
Homework Statement
A simple question about the topic diagonal matrix and diagonalization.
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A quick and simple question. I just realized that this has been posted in the wrong section, but ill give it a try anyway. Does anyone know if it's possible to diagonalize a hollow matrix? What i mean by a hollow matrix is a matrix with only zero entries along the diagonal.

Last edited:
Why not? Try diagonalizing$$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$and see what you get.

Sometimes. It depends on the matrix.
\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix}
is obviously diagonalizable.

## 1. What is a hollow matrix?

A hollow matrix is a matrix that contains mostly zero entries, with only a few non-zero entries. These non-zero entries are usually located along the diagonal of the matrix.

## 2. What is diagonalization of a matrix?

Diagonalization is a process of finding a diagonal matrix that is similar to the given matrix. This means that the two matrices have the same eigenvalues and eigenvectors.

## 3. Why is diagonalization important?

Diagonalization is important because it simplifies calculations involving matrices. It also allows us to easily find powers of matrices and solve systems of linear equations.

## 4. How do you diagonalize a matrix?

To diagonalize a matrix, we need to find its eigenvalues and eigenvectors. Then, we can use these eigenvectors to form a diagonal matrix and find the matrix similarity transformation that will convert the given matrix into its diagonal form.

## 5. Can all matrices be diagonalized?

No, not all matrices can be diagonalized. A matrix can only be diagonalized if it has a full set of linearly independent eigenvectors. If the matrix does not have enough eigenvectors, it cannot be diagonalized.

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