# Nearest block diagonal matrix to a given matrix

• cpp6f
In summary, the conversation discusses the problem of reducing a matrix to block diagonal form and the challenge of finding the closest block diagonal matrix to the original one. This problem arises in a quantum chemistry manuscript and the speaker is looking for a solution, but ultimately figures it out on their own.
cpp6f
Suppose I have a matrix that I want to reduce to block diagonal form. Obviously, the block diagonal form is not unique as each of the diagonal blocks is defined only to within a unitary rotation. So I want to find the block diagonal matrix that is closest to the original matrix in terms of the Frobenius norm.

This problem arises in a quantum chemistry manuscript I am putting together. If anyone can point me to a solution for this, I would be more than happy to acknowledge them in the manuscript.

Thanks!

Okay, I figured it out myself. After thinking about it for a week, I solve it right after posting!

## 1. What is a block diagonal matrix?

A block diagonal matrix is a square matrix that is composed of smaller square matrices, called blocks, along the main diagonal. The off-diagonal elements of the matrix are all zeros.

## 2. How do you find the nearest block diagonal matrix to a given matrix?

The nearest block diagonal matrix to a given matrix can be found by using a mathematical technique called matrix decomposition. This involves breaking down the given matrix into its constituent blocks and rearranging them to form a block diagonal matrix.

## 3. Can a non-square matrix have a nearest block diagonal matrix?

No, a non-square matrix cannot have a nearest block diagonal matrix. This is because a block diagonal matrix must have the same number of rows and columns, whereas a non-square matrix has a different number of rows and columns.

## 4. What are the applications of finding the nearest block diagonal matrix?

Finding the nearest block diagonal matrix is useful in data compression, image processing, and solving systems of linear equations. It can also be used to simplify and speed up matrix operations, as well as reduce the complexity of certain algorithms.

## 5. Can a given matrix have more than one nearest block diagonal matrix?

Yes, a given matrix can have multiple nearest block diagonal matrices. This is because there can be multiple ways to decompose a matrix into blocks and arrange them along the diagonal. However, the distance between the given matrix and each of its nearest block diagonal matrices will be the same.

Replies
2
Views
1K
Replies
1
Views
5K
Replies
8
Views
1K
Replies
7
Views
2K
Replies
4
Views
5K
Replies
4
Views
3K
Replies
4
Views
12K
Replies
1
Views
1K
Replies
156
Views
9K
Replies
8
Views
29K