In this tutorial, we will look at how to check if a numpy matrix (a 2d numpy array) is a symmetric matrix or not with the help of some examples.

### What is a symmetric matrix?

A matrix is said to be symmetric if it is equal to its transpose. That is, the matrix and its transpose are the same. The following image shows a symmetric matrix.

You can see that if you take the transpose of the above matrix, you’ll get the same matrix as above.

## How to check if a matrix is symmetric in Numpy?

To check if a matrix is symmetric, compare the matrix to its transpose, you can use the `.T`

property or the `numpy.transpose()`

function to get the transpose of the original matrix. You can use the `numpy.array_equal()`

method to compare the two matrices for equality.

import numpy as np # check if the matrix ar is symmetric np.array_equal(ar, ar.T)

In the above syntax, we’re basically checking if matric, `ar`

and its transpose, `ar.T`

are equal or not.

There are other methods as well –

- You can use the
`numpy.allclose()`

method to compare the two matrices for equality. Note that the`numpy.allclose()`

function uses a tolerance parameter to determine how close two values need to be to be considered equal. By default, the tolerance is set to 1e-05, which means that two values are considered equal if they are within 0.00001 of each other. You can adjust this tolerance by passing a different value to the`rtol`

(relative tolerance) or`atol`

(absolute tolerance) parameters of`numpy.allclose()`

.

Let’s now look at some examples of using the above syntax –

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### Example 1 – Using the `.T`

attribute and `numpy.array_equal()`

Let’s create a square matrix and check if it is symmetric. For this, we’ll take the following steps –

- Get the transpose of the matrix using
`.T`

attribute. - Compare the original matrix to its transpose for equality using the
`numpy.array_equal()`

method.

import numpy as np # create a symmetric matrix ar = np.array([[1, 2, 3], [2, 4, 5], [3, 5, 6]]) # check if the matrix ar is symmetric print(np.array_equal(ar, ar.T))

Output:

True

Here, we create a 3×3 symmetric matrix and checked if it’s symmetric or not. We get `True`

as the output indicating the array `ar`

is a symmetric matrix.

Let’s look at an example where the matrix is not a symmetric matrix.

import numpy as np # create a matrix ar = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) # check if the matrix ar is symmetric print(np.array_equal(ar, ar.T))

Output:

False

We get `False`

as the output which indicates that the matrix `ar`

is not symmetric.

### Example 2 – Using `.T`

attribute and `numpy.allclose()`

This method is similar to the above method, the only difference is that instead of `numpy.array_equal()`

function, we use the `numpy.allclose()`

function to compare the arrays. The following are the steps –

- Get the transpose of the matrix using the
`.T`

attribute. - Compare the original matrix to its transpose for equality using the
`numpy.allclose()`

method.

In this method, you can define how close two values need to be to be considered equal using the tolerance parameter.

Let’s take the same examples as above.

import numpy as np # create a symmetric matrix ar = np.array([[1, 2, 3], [2, 4, 5], [3, 5, 6]]) # check if the matrix ar is symmetric print(np.allclose(ar, ar.T))

Output:

True

import numpy as np # create a matrix ar = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) # check if the matrix ar is symmetric print(np.allclose(ar, ar.T))

Output:

False

We get the same results as above.

### Example 3 – Iterate through the matrix

Alternatively, you can iterate through the entire matrix and check whether each element satisfies the symmetric matrix property, `ar[i][j] == ar[j][i]`

.

import numpy as np # create a symmetric matrix ar = np.array([[1, 2, 3], [2, 4, 5], [3, 5, 6]]) # check if matrix is symmetric def is_matrix_symmetric(a): for i in range(len(a)): for j in range(len(a[i])): if a[i][j] == a[j][i]: continue else: return False return True # use the above function is_matrix_symmetric(ar)

Output:

True

You might also be interested in –

- How to check if a matrix is a square matrix in Numpy?
- How to check if a matrix is a diagonal matrix in Numpy?
- Numpy – Check if Matrix is a Lower Triangular Matrix
- Numpy – Check if Matrix is an Upper Triangular Matrix

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