conclude that the eigenvalues of $C_m^d$ are real, but not necessarily positive. For symmetric matrices, the singular values are, up to a sign, equal to the eigenvalues, so $$\sigma_i(C_m^d) = |\lambda_i(C_m^d)|.
  \label{4.2}$$ If the eigenvalues of the circulant $C_m^d$ are known, the spectrum of embedded $T_n^d$ can be bounded by the Cauchy interlace theorem for singular values, applied to $C_m^d$.

Let $C \in \mathbb{C}^{m \times n}$ be given, and let $C_\ell$ denote a submatrix of $C$ obtained by deleting a total of $\ell$ rows and/or $\ell$ columns of $C$. Then $$\sigma_k(C) \geq \sigma_k(C_\ell) \geq \sigma_{k+\ell}(C), \quad
    k = 1, \ldots, \min \{ m, n \},$$ where we set $\sigma_j(C) \equiv 0$ if $j > \min \{m, n \}$.

The proof can be found, for example, in [@Horn-Johnson-91 page 149].

If we delete the last $r$ rows and columns of $C_m^d$, we obtain $T_n^d$. The Cauchy interlace theorem will then give useful bounds for $\sigma_{\min}(T_n^d) = \lambda_{\min}(T_n^d)$, provided that $C_m^d$ is nonsingular. Moreover, if we delete more than $r$ last rows and columns of $C_m^d$, we obtain bounds for Töplitz matrices $T_k^d$, of order $k - d$, for $k \leq n$, $$\kappa_2(T_k^d) = \frac{\sigma_{\max}(T_k^d)}{\sigma_{\min}(T_k^d)}
  \leq \frac{\sigma_{\max}(C_m^d)}{\sigma_{\min}(C_m^d)}
  = \frac{1}{\min_j|\lambda_j(C_m^d)|}.
  \label{4.3}$$ Now we 