How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place?

$$ \hat{c}\hat{c}^\dagger+\hat{c}^\dagger\hat{c}=1 \\ c^2 = 0 \\ \overline{c}^2 = 0 \\ c\overline{c}+\overline{c}c = 0 \\ c\hat{c}+\hat{c} c = 0 \\ c\hat{c}^\dagger+\hat{c}^\dagger c = 0 \\ \overline{c}\hat{c}+\hat{c} \overline{c} = 0 \\ \overline{c}\hat{c}^\dagger+\hat{c}^\dagger \overline{c} = 0 \\ c\left|0\right>+\left|0\right>c = 0 \\ c\left|1\right>+\left|1\right>c = 0 \\ \overline{c}\left|0\right>+\left|0\right>\overline{c} = 0 \\ \overline{c}\left|1\right>+\left|1\right>\overline{c} = 0 \\ $$ It seems like it's impossible to represent both these operators and numbers as matrices or am I wrong? The anticommutation with states requires numbers to be antihermitian but it should be triangular to satisfy nilpotency, this only works for zero matrix.

In my case I need this also to work for two fermions with four states. I can represent operators as matrices and states as vectors as follows

$$ \left|0\right>=\{1,0,0,0\}\\ \left|\downarrow\right>=\{0,1,0,0\}\\ \left|\uparrow\right>=\{0,0,1,0\}\\ \left|\downarrow\uparrow\right>=\{0,0,0,1\}\\ \hat{c_\sigma} = \left(\array{0&\delta_{\sigma\downarrow}&\delta_{\sigma\uparrow}&0\\0&0&0&-\delta_{\sigma\uparrow}\\0&0&0&\delta_{\sigma\downarrow}\\0&0&0&0}\right)\\ c_\sigma = ?\\ \overline{c_\sigma}=? $$ So how should I represent these Grassmann numbers?

How one can simultaneously represent fermionic operators (denoted with hats) and its corresponding Grassmann variables (denoted without hats), so that all the anticommutation relations between them and also states would take place?

$$ \hat{c}\hat{c}^\dagger+\hat{c}^\dagger\hat{c}=1 \\ c^2 = 0 \\ \overline{c}^2 = 0 \\ c\overline{c}+\overline{c}c = 0 \\ c\hat{c}+\hat{c} c = 0 \\ c\hat{c}^\dagger+\hat{c}^\dagger c = 0 \\ \overline{c}\hat{c}+\hat{c} \overline{c} = 0 \\ \overline{c}\hat{c}^\dagger+\hat{c}^\dagger \overline{c} = 0 \\ c\left|0\right>-\left|0\right>c = 0 \\ c\left|1\right>+\left|1\right>c = 0 \\ \overline{c}\left|0\right>-\left|0\right>\overline{c} = 0 \\ \overline{c}\left|1\right>+\left|1\right>\overline{c} = 0. \\ $$ It seems like it's impossible to represent both these operators and Grassmann numbers as matrices or am I wrong? The anticommutation with states requires numbers to be antihermitian but it should be triangular to satisfy nilpotency, this only works for zero matrix.

In my case I need this also to work for two fermions with four states. I can represent operators as matrices and states as vectors as follows

$$ \left|0\right>=\{1,0,0,0\}\\ \left|\downarrow\right>=\{0,1,0,0\}\\ \left|\uparrow\right>=\{0,0,1,0\}\\ \left|\downarrow\uparrow\right>=\{0,0,0,1\}\\ \hat{c_\sigma} = \left(\array{0&\delta_{\sigma\downarrow}&\delta_{\sigma\uparrow}&0\\0&0&0&-\delta_{\sigma\uparrow}\\0&0&0&\delta_{\sigma\downarrow}\\0&0&0&0}\right)\\ c_\sigma = ?\\ \overline{c_\sigma}=? $$ So how should I represent these Grassmann numbers?

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place?

$$ \hat{c}\hat{c}^\dagger+\hat{c}^\dagger\hat{c}=1 \\ c^2 = 0 \\ \overline{c}^2 = 0 \\ c\overline{c}+\overline{c}c = 0 \\ c\hat{c}+\hat{c} c = 0 \\ c\hat{c}^\dagger+\hat{c}^\dagger c = 0 \\ \overline{c}\hat{c}+\hat{c} \overline{c} = 0 \\ \overline{c}\hat{c}^\dagger+\hat{c}^\dagger \overline{c} = 0 \\ c\left|0\right>+\left|0\right>c = 0 \\ c\left|1\right>+\left|1\right>c = 0 \\ \overline{c}\left|0\right>+\left|0\right>\overline{c} = 0 \\ \overline{c}\left|1\right>+\left|1\right>\overline{c} = 0 \\ $$ It seems like it's impossible to represent both these operators and numbers as matrices or am I wrong? The anticommutation with states requires numbers to be antihermitian but it should be triangular to satisfy nilpotency, this only works for zero matrix.

In my case I need this also to work for two fermions with four states.

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place?

$$ \hat{c}\hat{c}^\dagger+\hat{c}^\dagger\hat{c}=1 \\ c^2 = 0 \\ \overline{c}^2 = 0 \\ c\overline{c}+\overline{c}c = 0 \\ c\hat{c}+\hat{c} c = 0 \\ c\hat{c}^\dagger+\hat{c}^\dagger c = 0 \\ \overline{c}\hat{c}+\hat{c} \overline{c} = 0 \\ \overline{c}\hat{c}^\dagger+\hat{c}^\dagger \overline{c} = 0 \\ c\left|0\right>+\left|0\right>c = 0 \\ c\left|1\right>+\left|1\right>c = 0 \\ \overline{c}\left|0\right>+\left|0\right>\overline{c} = 0 \\ \overline{c}\left|1\right>+\left|1\right>\overline{c} = 0 \\ $$ It seems like it's impossible to represent both these operators and numbers as matrices or am I wrong? The anticommutation with states requires numbers to be antihermitian but it should be triangular to satisfy nilpotency, this only works for zero matrix.

In my case I need this also to work for two fermions with four states. I can represent operators as matrices and states as vectors as follows

$$ \left|0\right>=\{1,0,0,0\}\\ \left|\downarrow\right>=\{0,1,0,0\}\\ \left|\uparrow\right>=\{0,0,1,0\}\\ \left|\downarrow\uparrow\right>=\{0,0,0,1\}\\ \hat{c_\sigma} = \left(\array{0&\delta_{\sigma\downarrow}&\delta_{\sigma\uparrow}&0\\0&0&0&-\delta_{\sigma\uparrow}\\0&0&0&\delta_{\sigma\downarrow}\\0&0&0&0}\right)\\ c_\sigma = ?\\ \overline{c_\sigma}=? $$ So how should I represent these Grassmann numbers?