eye

→Tensor

eye(n: _int, m: _int | None = None, dtype: DTypeLike = None, device: DeviceLike = None, requires_grad: _bool = False)

source

Return a 2-D matrix with ones on the main diagonal and zeros elsewhere.

The returned matrix $E \in \mathbb{R}^{n \times m}$ is defined by the Kronecker delta:

E_{ij} = \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}

When $n = m$ this is the identity matrix $I_n$ , which satisfies $I_n A = A I_n = A$ for any $n \times n$ matrix $A$ . Rectangular variants ( $n \neq m$ ) arise naturally as the pseudo-identity in least-squares problems and in constructing projection matrices.

Parameters

nint

Number of rows.

mint

Number of columns. Defaults to n, yielding a square identity.

dtypelucid.dtype

Scalar data type. Defaults to the global default dtype (lucid.float32 unless overridden).

devicestr or lucid.device

Target device — "cpu" or "metal".

requires_gradbool

If True, downstream operations are tracked by autograd. Default: False.

Returns

Tensor

2-D tensor of shape (n, m) with $E_{ij} = \delta_{ij}$ .

Notes

The identity matrix is its own inverse and its own transpose: $I_n^{-1} = I_n^\top = I_n$ . Its eigenvalues are all $1$ , and it is simultaneously orthogonal, symmetric, idempotent ( $I^2 = I$ ), and unitary.

In deep learning, identity initialisations are used in recurrent networks (e.g. the IRNN, Le et al. 2015) to preserve gradient norms across time steps, exploiting the fact that the spectral radius of $I$ is exactly $1$ .

Examples

>>> import lucid
>>> lucid.eye(3)
Tensor([[1., 0., 0.],
        [0., 1., 0.],
        [0., 0., 1.]])
Rectangular variant:
>>> lucid.eye(2, 4)
Tensor([[1., 0., 0., 0.],
        [0., 1., 0., 0.]])
Verify $I A = A$:
>>> A = lucid.tensor([[1., 2.], [3., 4.]])
>>> (lucid.eye(2) @ A - A).abs().max()
Tensor(0.)