Editing Toeplitz matrix (section)

==Infinite Toeplitz matrix==
{{Main|Toeplitz operator}}
A bi-infinite Toeplitz matrix (i.e. entries indexed by <math>\mathbb Z\times\mathbb Z</math>) <math>A</math> induces a [[linear operator]] on <math>\ell^2</math>.

:<math>
A=\begin{bmatrix}
       & \vdots & \vdots & \vdots & \vdots \\
\cdots & a_0 & a_{-1} & a_{-2} & a_{-3} & \cdots \\
\cdots & a_1 & a_0 &  a_{-1} & a_{-2} & \cdots   \\
\cdots & a_2 & a_1 & a_0 & a_{-1} & \cdots  \\
\cdots & a_3 & a_2 & a_1 & a_0 & \cdots \\
       & \vdots  & \vdots & \vdots & \vdots
\end{bmatrix}.
 
</math>

The induced operator is [[bounded operator|bounded]] if and only if the coefficients of the Toeplitz matrix <math>A</math> are the Fourier coefficients of some [[Essential range|essentially bounded]] function <math>f</math>.

In such cases, <math>f</math> is called the '''symbol''' of the Toeplitz matrix <math>A</math>, and the spectral norm of the Toeplitz matrix <math>A</math> coincides with the <math>L^\infty</math> norm of its symbol. The [[mathematical proof|proof]] can be found as Theorem 1.1 of Böttcher and Grudsky.<ref>{{harvnb|Böttcher|Grudsky|2012}}</ref>