imate.sample_matrices.toeplitz_trace

imate.sample_matrices.toeplitz_trace(a, b, size, gram=False)

Compute the trace of the Toeplitz matrix using an analytic formula.

The Toeplitz matrix using the entries \(a\) and \(b\) refers to the matrix generated by imate.toeplitz().

Parameters:

afloat

The diagonal elements of the Toeplitz matrix.

bfloat

The upper off-diagonal elements of the Toeplitz matrix.

sizeint, default=20

Size of the square matrix.

grambool, default=False

If False, the matrix is assumed to be bi-diagonal Toeplitz. If True, the Gramian of the matrix is considered instead.

Returns:

tracefloat

Trace of Toeplitz matrix

See also

imate.toeplitz

imate.sample_matrices.toeplitz_logdet

imate.sample_matrices.toeplitz_traceinv

Notes

For the matrix \(\mathbf{A}\) given in imate.toeplitz(), the trace is computed by

\[\mathrm{trace}(\mathbf{A}) = n a,\]

where \(n\) is the size of the matrix. For the Gramian matrix, \(\mathbf{B} = \mathbf{A}^{\intercal} \mathbf{A}\) (when gram is set to True), the trace is

\[\mathrm{trace}(\mathbf{B}) = a^2 + (n-1)(a^2 + b^2).\]

Examples

>>> from imate.sample_matrices import toeplitz_trace
>>> a, b = 2, 3

>>> toeplitz_trace(a, b, size=6)
12

>>> toeplitz_trace(a, b, size=6, gram=True)
69