NumPy ndarray.trace() Method

NumPy ndarray.trace()

The numpy.ndarray.trace() method computes the sum of the diagonal elements of a NumPy array. It allows specifying an offset to shift the diagonal, as well as the axes to extract the diagonal from.

Syntax

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ndarray.trace(offset=0, axis1=0, axis2=1, dtype=None, out=None)

Parameters

Parameter	Type	Description
`offset`	int, optional	Specifies which diagonal to sum. Default is `0` (main diagonal). Positive values move the diagonal above, negative values below.
`axis1`	int, optional	The first axis of the 2D sub-array from which the diagonal is taken. Default is `0`.
`axis2`	int, optional	The second axis of the 2D sub-array from which the diagonal is taken. Default is `1`.
`dtype`	dtype, optional	Data type to use for summation. If `None`, it uses the array’s data type.
`out`	ndarray, optional	Alternative output array to store the result. Must match expected output shape.

Return Value

Returns a scalar value representing the sum of the specified diagonal elements.

Examples

1. Computing the Trace of a Square Matrix

In this example, we compute the trace of a 3×3 matrix, which is the sum of its main diagonal elements.

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import numpy as np

# Define a 3x3 matrix
arr = np.array([[1, 2, 3],
                [4, 5, 6],
                [7, 8, 9]])

# Compute the trace (sum of main diagonal elements: 1 + 5 + 9)
result = arr.trace()

print("Trace of the matrix:", result)

Output:

Trace of the matrix: 15

2. Using the `offset` Parameter

The offset parameter shifts the diagonal from which the sum is computed. A positive value selects diagonals above the main diagonal, while a negative value selects below.

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import numpy as np

# Define a 4x4 matrix
arr = np.array([[1, 2, 3, 4],
                [5, 6, 7, 8],
                [9, 10, 11, 12],
                [13, 14, 15, 16]])

# Compute the trace of the diagonal above the main diagonal (offset=1)
result_offset1 = arr.trace(offset=1)

# Compute the trace of the diagonal below the main diagonal (offset=-1)
result_offset_neg1 = arr.trace(offset=-1)

print("Trace with offset=1:", result_offset1)   # 2 + 7 + 12 = 21
print("Trace with offset=-1:", result_offset_neg1)  # 5 + 10 + 15 = 30

Output:

Trace with offset=1: 21
Trace with offset=-1: 30

3. Specifying Custom Axes for the Diagonal

The axis1 and axis2 parameters allow computing the trace along different axes in multi-dimensional arrays.

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import numpy as np

# Define a 3D array (2x3x3)
arr = np.array([[[1, 2, 3], 
                 [4, 5, 6], 
                 [7, 8, 9]],

                [[10, 11, 12], 
                 [13, 14, 15], 
                 [16, 17, 18]]])

# Compute the trace along the specified axes (axis1=1, axis2=2)
result = arr.trace(axis1=1, axis2=2)

print("Trace along custom axes:\n", result)

Output:

[15 42]

Here, the function extracts the diagonal from each 3×3 sub-array and sums them, returning one value per 2D slice.

4. Changing Data Type with `dtype`

By using the dtype parameter, we can specify the precision used for summation.

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import numpy as np

# Define a float matrix
arr = np.array([[1.1, 2.2, 3.3],
                [4.4, 5.5, 6.6],
                [7.7, 8.8, 9.9]])

# Compute trace with integer dtype
result = arr.trace(dtype=int)

print("Trace with integer dtype:", result)

Output:

Trace with integer dtype: 15

Here, the floating-point values are cast to integers before summation, truncating the decimal part.

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NumPy ndarray.trace() Method