How to solve this equation of n-order matrix?

Question

Question:

Let i be an n-level identity matrix and j be an n-level matrix with all elements of 1.

Let n-order matrix m be

$M=\left(\begin{array}{ccccc}k & \lambda & \lambda & \cdots & \lambda \\ \lambda & k & \lambda & \cdots & \lambda \\ \vdots & \vdots & \vdots & & \vdots \\ \lambda & \lambda & \lambda & \cdots & k\end{array}\right)$

How to express m in the form of x i + y j, where x and y are coefficients.

My code:

When n is a definite positive integer, I can successfully run the code to solve x and y.

For example, n==5:

Clear["Global`*"];

m = DiagonalMatrix[Table[k - \!\(TraditionalForm\`\[Lambda]\), {5}]] +
   ConstantArray[\!\(TraditionalForm\`\[Lambda]\), {5, 5}]

i = IdentityMatrix[5]

j = ConstantArray[1, {5, 5}]

Reduce[x*i + y*j == m, {x, y}]

$\mathrm{x}=\mathrm{k}-\lambda \operatorname{\& \& } \mathrm{y}==\lambda$

But when n is an uncertain positive integer (just a symbolic variable), the code fails.

Clear["Global`*"];

$Assumptions = (m | i | j) \[Element] Matrices[{n, n}];

m[n_Integer?Positive] := 
 DiagonalMatrix[Table[k - \!\(TraditionalForm\`\[Lambda]\), {n}]] + 
  ConstantArray[\!\(TraditionalForm\`\[Lambda]\), {n, n}]

i[n_Integer?Positive] := IdentityMatrix[n];

j[n_Integer?Positive] := ConstantArray[1, {n, n}];

Reduce[x*i[n] + y*j[n] == m[n], {x, y}]

$\left(j[n] \neq 0 \& \& y=\frac{-x i[n]+m[n]}{j[n]}\right)||\left(j[n]=0 \& \& i[n] \neq 0 \& \& x=\frac{m[n]}{i[n]}\right)||(m[n]=0 \& \& j[n]=0 \& \& i[n]=0)$

I wonder if MMA can solve this problem?

Answer 1

Does this work for you?

ident[i_, j_] := KroneckerDelta[i, j]
jmat[i_, j_] := 1
mmat[i_, j_] := (k - \[Lambda])  KroneckerDelta[i, j] + \[Lambda]

then:

res = Assuming[
 {i \[Element] Integers && j \[Element] Integers},
 Reduce[x ident[i, j] + y jmat[i, j] == mmat[i, j], {x, y}, Reals]
 ]

gives me:

and,

FullSimplify[res]

gives me