In this article we will discuss the steps and intuition for creating the diagonal matrix and show examples using Python.

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## Introduction

The diagonal matrix ($$D$$) is most often seen in linear algebra expressions that involve the identity matrices.

To continue following this tutorial we will need the following Python library: numpy.

If you don’t have them installed, please open “Command Prompt” (on Windows) and install them using the following code:

pip install numpy


## Diagonal matrix explained

We are already familiar with the identity matrix and some of its properties, and it actually is a special case of a diagonal matrix.

A diagonal matrix is a matrix (usually a square matrix of order $$n$$) filled with values on the main diagonal and zeros everywhere else.

Here are a few examples:

$$D_1 = \begin{bmatrix} 3 \end{bmatrix}$$

$$D_2 = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}$$

$$D_3 = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 5 \end{bmatrix}$$

and so on for the larger dimensions.

Graphically, the $$D_2$$ matrix simply represents the scaled base vectors:

$$\vec{d}_1 = (3, 0)$$

$$\vec{d}_2 = (0, 2)$$

There are usually two main use cases for the diagonal matrix:

1. Creating a diagonal matrix – implies taking a vector and creating a matrix where the values from a vector become the main diagonal of a matrix.
2. Extracting a diagonal from a matrix – implies finding the main diagonal in a given matrix.

## Create diagonal matrix using Python

In order to create a diagonal matrix using Python we will use the numpy library. And the first step will be to import it:

import numpy as np


Numpy has a lot of useful functions, and for this operation we will use the diag() function. This function is particularly interesting, because if we pass a 1-D array into it, it will return a 2-D array (or a matrix) with the vector values on its $$k$$-th diagonal ($$k$$=0 for the main diagonal).

Now let’s create some 1-D array (or a vector):

v = [3, 2, 5]


and create a matrix with $$v$$ on the main diagonal:

D = np.diag(v)

print(D)


And you should get:

[[3 0 0]
[0 2 0]
[0 0 5]]

which is a diagonal matrix with values on the main diagonal and zeros everywhere else.

## Extract diagonal from matrix using Python

In order to extract a diagonal from a matrix using Python we will use the numpy library. And the first step will be to import it:

import numpy as np


Numpy has a lot of useful functions, and for this operation we will use the diag() function. This function is particularly interesting, because if we pass a 2-D array into it, it will return its $$k$$-th diagonal ($$k$$=0 for the main diagonal).

Now let’s create some 3×3 matrix:

A = np.array([[3, 1, 4],
[0, 2, 7],
[8, 9, 5]])


and extract it’s main diagonal:

d = np.diag(A)

print(d)


And you should get:

[3 2 5]

which are exactly the values on the main diagonal of the matrix.

## Conclusion

In this article we discussed the steps and intuition for creating the diagonal matrix, as well as extracting a diagonal from a matrix using Python.

Feel free to leave comments below if you have any questions or have suggestions for some edits and check out more of my Linear Algebra articles.

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