deck

A = \begin{bmatrix} a_{0,0} & a_{0,1} &&&&&& a_{0,7}\\ a_{1,0} & a_{1,1} &&&&& a_{1,6}\\ && a_{2,2} & a_{2,3} && a_{2,5} \\ && a_{3,2} & a_{3,3} & a_{3,4}\\ &&& a_{4,3} & a_{4,4} & a_{4,5}\\ && a_{5,2} && a_{5,5} & a_{5,5} & \\ & a_{6,1}&&&&& a_{6,6} & a_{6,7} \\ a_{7,0} &&&&&& a_{7,6} & a_{7,7}\\ \end{bmatrix}

C = \begin{bmatrix} c_{0}\\ c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ c_{5}\\ c_{6}\\ c_{7}\\ \end{bmatrix}

A = \begin{bmatrix} a_{0,0} & a_{0,1} & a_{0,15}\\ a_{1,0} & a_{1,1} & a_{1,14}\\ a_{2,2} & a_{2,3} & a_{2,13} \\ a_{3,2} & a_{3,3} & a_{3,12}\\ a_{4,4} & a_{4,5} & a_{4,11}\\ a_{5,4} & a_{5,5} & a_{5,10}\\ a_{6,6} & a_{6,7} & a_{6,9}\\ a_{7,6} & a_{7,7} & a_{7,8}\\ a_{8,7} & a_{8,8} & a_{8,9}\\ a_{9,6} & a_{9,8} & a_{9.9} \\ a_{10,5} & a_{10,10} & a_{10,11} \\ a_{11,4} & a_{11,10} & a_{11,11} \\ a_{12,3} & a_{12,12} & a_{12,13} \\ a_{13,2} & a_{13,12} & a_{13,13} \\ a_{14,1} & a_{14,14} & a_{14,15}\\ a_{15,0} & a_{15,14} & a_{15,15} \\ \end{bmatrix}

C = \begin{bmatrix} c_{0}\\ c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ c_{5}\\ c_{6}\\ c_{7}\\ c_{8}\\ c_{9}\\ c_{10}\\ c_{11}\\ c_{12}\\ c_{13}\\ c_{14}\\ c_{15}\\ \end{bmatrix}

$A = \left[ \begin{smallmatrix}a_{0,0} \ & a_{0,1} & & & & & & & & & & & & & & a_{0,15}\\a_{1,0} & a_{1,1} & & & & & & & & & & & & & a_{1,14}\\& & a_{2,2} & a_{2,3} & & & & & & & & & & a_{2,13} \\& & a_{3,2} & a_{3,3} & & & & & & & & & a_{3,12}\\& & & & a_{4,4} & a_{4,5} & & & & & & a_{4,11}\\& & & & a_{5,4} & a_{5,5} & & & & & a_{5,10}\\& & & & & & a_{6,6} & a_{6,7} & & a_{6,9}\\& & & & & & a_{7,6} & a_{7,7} & a_{7,8}\\& & & & & & & a_{8,7}& a_{8,8} & a_{8,9}\\& & & & & & a_{9,6}& & a_{9,8} & a_{9.9} &\\& & & & & a_{10,5} & & & & & a_{10,10} & a_{10,11} \\& & & & a_{11,4} & & & & & & a_{11,10} & a_{11,11} & \\& & & a_{12,3} & & & & & & & & & a_{12,12} & a_{12,13} \\& & a_{13,2} & & & & & & & & & & a_{13,12} & a_{13,13} &\\& a_{14,1}& & & & & & & & & & & & & a_{14,14} & a_{14,15}\\a_{15,0}& & & & & & & & & & & & & & a_{15,14} & a_{15,15} \\\end{smallmatrix} \right]$

$C = \left[ \begin{smallmatrix} c_{0}\\ c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ c_{5}\\ c_{6}\\ c_{7}\\ c_{8}\\ c_{9}\\ c_{10}\\ c_{11}\\ c_{12}\\ c_{13}\\ c_{14}\\ c_{15}\\ \end{smallmatrix}\right] $

A = \begin{bmatrix} a_{0,0} & a_{0,1} & & & & & & & & & & & & & & a_{0,15}\\ a_{1,0} & a_{1,1} & & & & & & & & & & & & & a_{1,14}\\ & & a_{2,2} & a_{2,3} & & & & & & & & & & a_{2,13} \\ & & a_{3,2} & a_{3,3} & & & & & & & & & a_{3,12}\\ & & & & a_{4,4} & a_{4,5} & & & & & & a_{4,11}\\ & & & & a_{5,4} & a_{5,5} & & & & & a_{5,10}\\ & & & & & & a_{6,6} & a_{6,7} & & a_{6,9}\\ & & & & & & a_{7,6} & a_{7,7} & a_{7,8}\\ & & & & & & & a_{8,7}& a_{8,8} & a_{8,9}\\ & & & & & & a_{9,6}& & a_{9,8} & a_{9.9} &\\ & & & & & a_{10,5} & & & & & a_{10,10} & a_{10,11} \\ & & & & a_{11,4} & & & & & & a_{11,10} & a_{11,11} & \\ & & & a_{12,3} & & & & & & & & & a_{12,12} & a_{12,13} \\ & & a_{13,2} & & & & & & & & & & a_{13,12} & a_{13,13} &\\ & a_{14,1}& & & & & & & & & & & & & a_{14,14} & a_{14,15}\\ a_{15,0}& & & & & & & & & & & & & & a_{15,14} & a_{15,15} \\ \end{bmatrix}

C = \begin{bmatrix} c_{0}\\ c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ c_{5}\\ c_{6}\\ c_{7}\\ c_{8}\\ c_{9}\\ c_{10}\\ c_{11}\\ c_{12}\\ c_{13}\\ c_{14}\\ c_{15}\\ \end{bmatrix}

\begin{bmatrix} a_{0,0} & a_{0,1} & a_{0,7}\\ a_{1,0} & a_{1,1} & a_{1,6}\\ a_{2,2} & a_{2,3} & a_{2,5} \\ a_{3,2} & a_{3,3} & a_{3,4}\\ a_{4,3} & a_{4,4} & a_{4,5}\\ a_{5,2} & a_{5,5} & a_{5,5} \\ a_{6,1} & a_{6,6} & a_{6,7} \\ a_{7,0} & a_{7,6} & a_{7,7}\\ \end{bmatrix}

\begin{bmatrix} c_{0}\\ c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\\ c_{5}\\ c_{6}\\ c_{7}\\ \end{bmatrix}

P = \begin{bmatrix} & p_{1} \\ p_{0} \\ & & & p_{3} &\\ & & p_{2} &\\ & & & & & p_{5} &\\ & & & & p_{4} &\\ & & & & & & & p_{7} &\\ & & & & & & p_{6} &\\ & & & & & & & & & p_{9} &\\ & & & & & & & & p_{8} &\\ & & & & & & & & & & & p_{11} &\\ & & & & & & & & & & p_{10} &\\ & & & & & & & & & & & & & p_{13} &\\ & & & & & & & & & & & & p_{12} &\\ & & & & & & & & & & & & & & & p_{15} \\ & & & & & & & & & & & & & & p_{14} &\\ \end{bmatrix}

P = \begin{bmatrix} & & & p_{3} &\\ & & p_{2} &\\ & p_{1} & &\\ p_{0} & & & \\ & & & & & & & p_{7} &\\ & & & & & & p_{6} &\\ & & & & & p_{5} &\\ & & & & p_{4} &\\ & & & & & & & & & & & p_{11} &\\ & & & & & & & & & & p_{10} &\\ & & & & & & & & & p_{9} &\\ & & & & & & & & p_{8} &\\ & & & & & & & & & & & & & & & p_{15}\\ & & & & & & & & & & & & & & p_{14} &\\ & & & & & & & & & & & & & p_{13} &\\ & & & & & & & & & & & & p_{12} &\\ \end{bmatrix}

P = \begin{bmatrix} & & & & p_{4} &\\ & & & & & p_{5} &\\ & & & & & & p_{6} &\\ & & & & & & & p_{7} &\\ p_{0} & & & \\ & p_{1} & &\\ & & p_{2} &\\ & & & p_{3} &\\ & & & & & & & & & & & & p_{12} &\\ & & & & & & & & & & & & & p_{13} &\\ & & & & & & & & & & & & & & p_{14} &\\ & & & & & & & & & & & & & & & p_{15} \\ & & & & & & & & p_{8} &\\ & & & & & & & & & p_{9} &\\ & & & & & & & & & & p_{10} &\\ & & & & & & & & & & & p_{11} &\\ \end{bmatrix}

P = \begin{bmatrix} & & & & & p_{5} &\\ & & & & p_{4} &\\ & & & & & & & p_{7} &\\ & & & & & & p_{6} &\\ & p_{1} & &\\ p_{0} & & & \\ & & & p_{3} &\\ & & p_{2} &\\ & & & & & & & & & & & & & p_{13} &\\ & & & & & & & & & & & & p_{12} &\\ & & & & & & & & & & & & & & & p_{15} \\ & & & & & & & & & & & & & & p_{14} &\\ & & & & & & & & & p_{9} &\\ & & & & & & & & p_{8} &\\ & & & & & & & & & & & p_{11} &\\ & & & & & & & & & & p_{10} &\\ \end{bmatrix}

H = \begin{bmatrix} & & & & & & & & & & & & & & & p_{15} \\ & & & & & & & & & & & & & & p_{14} &\\ & & & & & & & & & & & & & p_{13} &\\ & & & & & & & & & & & & p_{12} &\\ & & & & & & & & & & & p_{11} &\\ & & & & & & & & & & p_{10} &\\ & & & & & & & & & p_{9} &\\ & & & & & & & & p_{8} &\\ & & & & & & & p_{7} &\\ & & & & & & p_{6} &\\ & & & & & p_{5} &\\ & & & & p_{4} &\\ & & & p_{3} &\\ & & p_{2} &\\ & p_{1} & &\\ p_{0} & & & \\ \end{bmatrix}

$a_1$

$a_2$

$a_3$

$\vdots$

$a_n$

$b_1$

$b_2$

$b_3$

$b_n$

$\vdots$

$a_1 - b_1$

$a_2 - b_2$

$a_3 - b_3$

$\vdots$

$a_n-b_n$

$$\rho = \begin{bmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ddots & \vdots\\ a_{1n} & \ldots & a_{nn} \end{bmatrix} = VDV^\dagger$$