numpy.trace(a, offset=0, axis1=0, axis2=1, dtype=None, out=None)[source]

Return the sum along diagonals of the array.

If a is 2-D, the sum along its diagonal with the given offset is returned, i.e., the sum of elements a[i,i+offset] for all i.

If a has more than two dimensions, then the axes specified by axis1 and axis2 are used to determine the 2-D sub-arrays whose traces are returned. The shape of the resulting array is the same as that of a with axis1 and axis2 removed.


Input array, from which the diagonals are taken.

offsetint, optional

Offset of the diagonal from the main diagonal. Can be both positive and negative. Defaults to 0.

axis1, axis2int, optional

Axes to be used as the first and second axis of the 2-D sub-arrays from which the diagonals should be taken. Defaults are the first two axes of a.

dtypedtype, optional

Determines the data-type of the returned array and of the accumulator where the elements are summed. If dtype has the value None and a is of integer type of precision less than the default integer precision, then the default integer precision is used. Otherwise, the precision is the same as that of a.

outndarray, optional

Array into which the output is placed. Its type is preserved and it must be of the right shape to hold the output.


If a is 2-D, the sum along the diagonal is returned. If a has larger dimensions, then an array of sums along diagonals is returned.

See also

diag, diagonal, diagflat


>>> np.trace(np.eye(3))
>>> a = np.arange(8).reshape((2,2,2))
>>> np.trace(a)
array([6, 8])
>>> a = np.arange(24).reshape((2,2,2,3))
>>> np.trace(a).shape
(2, 3)