MatrixForm with zeroes formatted in gray

Question

I would like to define a version of MatrixForm that prints entries equal to zero in gray. The definition

matrixform[X_] := MatrixForm[X /. entry_ /; entry == 0 -> Style[0, LightGray]]

produces an output that looks how I'd want, but it actually changes the entries, rather than just the way they are displayed, so one cannot just use the output of my matrixform via % in algebraic manipulations.

Ideally I'm looking for an answer that works for sparse arrays too, without the conversion to a normal array: at least, I'd like to be able to work with the output without having to convert it back to a sparse array.

Note that I don't want just all zeroes in my notebook to be formatted in gray (so I don't think I can use Format), and I don't want matrices to be formatted as above automatically either.

---

Edit. Here is an example of what I'm hoping to achieve:

A = SparseArray[{1,1}->x,{2,2}]
(* Out: SparseArray[..., Specified elements: 1, Dimensions: {2,2}] as usual *) 

% // matrixform
(* Out: MatrixForm of A with 0s shown in gray *)

2 %
(* Out: SparseArray[..., Specified elements: 1, Dimensions: {2,2}] as usual *)

% // matrixform
(* Out: MatrixForm of 2*A with 0s shown in gray *)

Answer 1

To get the % behavior you want, you need to create a wrapper, that is, a function that does not evaluate, but creates a nice format when it is rendered. The usual way to do this is to use MakeBoxes. Here is a version that should target only matrix entries, and also avoid evaluation leaks:

matrixForm /: MakeBoxes[matrixForm[x_List], StandardForm] := Replace[
    Replace[Hold[x], 0->Style[0, Red], {3}],
    Hold[z_] :> MakeBoxes[MatrixForm[z]]
]

Note that I used Red instead of LightGray to improve visualization for this answer. Now that matrixForm is a wrapper, you can add it to $OutputForms as suggested in (51898).

Unprotect[$OutputForms];
AppendTo[$OutputForms, matrixForm];
Protect[$OutputForms];

Example:

matrixForm[{{f[0], 0}, {1, 0}}]

And using %:

%

{{f[0], 0}, {1, 0}}

For SparseArray objects, you could add the following format, but note that it relies on being able to reconstruct a SparseArray from its FullForm, updated to work in M13 as well (note that being able to reconstruct a sparse array in this way may change in the future):

matrixForm /: MakeBoxes[matrixForm @ s_SparseArray, StandardForm] := With[
    {
    new = SparseArray[
        Automatic, 
        Dimensions[s], 
        Style[s["Background"], Red],
        {1, {s["RowPointers"], s["ColumnIndices"]}, s["NonzeroValues"]}]
    },
    MakeBoxes[
        MatrixForm @ new,
        StandardForm
    ]
]

Example:

matrixForm[IdentityMatrix[3, SparseArray]]

Check %:

%

SparseArray[Automatic, {3, 3}, 0, { 1, {{0, 1, 2, 3}, {{1}, {2}, {3}}}, {1, 1, 1}}]

Answer 2

SeedRandom[0]

(m = RandomInteger[{0, 2}, {4, 4}]) // MatrixForm

Note that by using parentheses, the definition of m does not include the wrapper MatrixForm

matrixForm[x_List] := ((x /. 0 :> Style[0, LightGray]) // MatrixForm)

m // matrixForm

The wrappers, MatrixForm or matrixForm are only for display. If you want to use the matrix, use m. Any subsequent operation is done on m and the display is formatted using the wrapper.

(m2 = Transpose@m) // matrixForm

m == (m2 // Transpose)

(* True *)