How to compute eigenvalues of linear function (not matrix)?

How to compute eigenvalues of a known linear function? In Julia, there is a package to compute the matrix representation of given function, then we can compute eigenvalues of the final matrix.

And in MATLAB, function eigs can also directly compute eigenvalues of a linear function.

     d = eigs(Afun,n,___) specifies a function handle Afun instead of a matrix. The second input n gives the size of matrix A used in Afun. You can optionally specify B, k, sigma, opts, or name-value pairs as additional input arguments. 

enter image description here So how to realize them in Mma? This is my approach:

LinearMapMatrix[map_Association, dimension_] :=   Module[{vectorin, linearfunction, transformmatrix},    vectorin = Normal[map][[1, 1]]; linearfunction = map[vectorin];    transformmatrix =     Transpose[ParallelMap[linearfunction, IdentityMatrix[dimension]]]   ] 

But it costs a lot of Ram. If I want a matrix representation with huge dimension, this function would burn down the running notebook. Are there some better methods?