Defining a function for the construction of a bordered hessian

Question

I think in Mathematica the omission of a function that can create a bordered matrix. This code works

m = {{a, b}, {c, d}};
k = {α, β}
l1 = Insert[ k, 0, 1];
c1 = Insert[Transpose[m], k, 1] // Transpose;
m1 = Insert[c1, l1, 1];
m1 // MatrixForm

and gives

$\begin{pmatrix} 0 & \alpha & \beta\\ \alpha & a & b\\ \beta & c & d \end{pmatrix}$

as expected.

So I have tried to design a function. But obviously it doesn't work

bordhess[mat_?MatrixQ[mat], vec_?VecQ[vec] ] := 
 Block[{x = mat, y = vec},
  l1 = Insert[ y, 0, 1];
  c1 = Insert[Transpose[x], k, 1] // Transpose;
  Insert[c1, l1, 1]
  ]

and even If I have been able to create such a function, I wonder how to generalize the problem to many vectors.

Answer 1

Also,

borderF = ArrayReshape[{0, #, Transpose[{##}]}, 1 + Dimensions@#2] &;

mat = Array[Subscript[a, ##] &, {4, 4}];
multipliers = Array[Subscript[λ, #] &, 4];
borderF[multipliers, mat] // MatrixForm

Answer 2

You can do this with ArrayFlatten. Here is a generalization to one or more vectors -- I'm not sure if it's what you're looking for:

bordhess[mat_?MatrixQ, mat1_?MatrixQ] := ArrayFlatten[{{0, mat1}, {Transpose@mat1, mat}}]

bordhess[mat_?MatrixQ, vec_?VectorQ] := bordhess[mat, {vec}]

Then you have

m = {{a, b}, {c, d}};
k = {α, β};
j = {γ, δ};

bordhess[m, k] // MatrixForm

bordhess[m, {k, j}] // MatrixForm

Answer 3

Here's an answer to the title question, about constructing a bordered Hessian, in case someone come looking for answer to it. It comes directly from calculus, instead of playing with matrices.

Basically thus:

D[f + λ.g, {{λ1, λ2,..., x1, x2,...}, 2}]

where

λ = {λ1, λ2,...}

are the Lagrange multipliers and

x = {x1, x2,...}

are the variables of f and g.

Example:

ClearAll[bH];
bH[f_, g_List, x_List, param_: K] := bH[f, g, x, Array[param, Length@g]];
bH[f_, g_List, x_List, λ_List] := D[f + λ.g, {Join[λ, x], 2}];

bH[w^4 x + x^3 y + y^2 z + z^4,
 {w^2 + x^2 + y^2 + z^2, w^2 + 2 x^2 + 3 y^2 + 4 z^2},
 {w, x, y, z}, {λ, μ}] // MatrixForm