I have a question about the computation of the norm of a block matrices.
The square block matrix $T_{n}$ is defined as:
\begin{bmatrix} A_{1,1} &. &.&&A_{1,n} \\ A_{2,1} &. &.&.&A_{2,n}\\ . &. &.&.&.\\ . &. &.&.&.\\ . &. &.&.&.\\ A_{n,1} &. &.&.&A_{n,n}\\ \end{bmatrix} where $A_{i,j}$ is a $c \times c$ Toeplitz matrix given as \begin{bmatrix} a_{0} &a_{1} &.&.&a_{c} \\ a_{-1} &a_{0} &a_{1}&.&.\\ . &. &.&.&.\\ . &. &.&.&a_{1}\\ a_{-c} &. &.&a_{-1}&a_{0}\\ \end{bmatrix}
I would like to compute the following norm
$||T_{n}||^{2}=\underset{x}{\max}(x'T_{n}T'_{n}x)/(x'x)$.
In the case of a simple $n \times n$ Toeplitz matrix, it is easy and gives
$||T_{n}||^{2}=\underset{x}{\max}(x'T_{n}T'_{n}x)/(x'x)= \sum \limits_{\underset{}{k=0}}^{n-1}\sum \limits_{\underset{}{j=0}}^{n-1} t_{k-j}x'x /(\sum \limits_{\underset{}{k=0}}^{n-1} |x_{k}|^{2})$
Any ideas or suggestions are welcomed.