# scipy.linalg.dft¶

scipy.linalg.dft(n, scale=None)[source]

Discrete Fourier transform matrix.

Create the matrix that computes the discrete Fourier transform of a sequence [1]. The nth primitive root of unity used to generate the matrix is exp(-2*pi*i/n), where i = sqrt(-1).

Parameters
nint

Size the matrix to create.

scalestr, optional

Must be None, ‘sqrtn’, or ‘n’. If scale is ‘sqrtn’, the matrix is divided by sqrt(n). If scale is ‘n’, the matrix is divided by n. If scale is None (the default), the matrix is not normalized, and the return value is simply the Vandermonde matrix of the roots of unity.

Returns
m(n, n) ndarray

The DFT matrix.

Notes

When scale is None, multiplying a vector by the matrix returned by `dft` is mathematically equivalent to (but much less efficient than) the calculation performed by `scipy.fft.fft`.

New in version 0.14.0.

References

1

“DFT matrix”, https://en.wikipedia.org/wiki/DFT_matrix

Examples

```>>> from scipy.linalg import dft
>>> np.set_printoptions(precision=2, suppress=True)  # for compact output
>>> m = dft(5)
>>> m
array([[ 1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ],
[ 1.  +0.j  ,  0.31-0.95j, -0.81-0.59j, -0.81+0.59j,  0.31+0.95j],
[ 1.  +0.j  , -0.81-0.59j,  0.31+0.95j,  0.31-0.95j, -0.81+0.59j],
[ 1.  +0.j  , -0.81+0.59j,  0.31-0.95j,  0.31+0.95j, -0.81-0.59j],
[ 1.  +0.j  ,  0.31+0.95j, -0.81+0.59j, -0.81-0.59j,  0.31-0.95j]])
>>> x = np.array([1, 2, 3, 0, 3])
>>> m @ x  # Compute the DFT of x
array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])
```

Verify that `m @ x` is the same as `fft(x)`.

```>>> from scipy.fft import fft
>>> fft(x)     # Same result as m @ x
array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])
```