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lanxiyu

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Registered: 2022-05-10

Posts: 55

Mathematica code for matrix calculus

(* Addition *)
D[f[x] + g[x], x] == f'[x] + g'[x]
D[f[x] + g[x], {x, 2}] == f''[x] + g''[x]

(* Subtraction *)
D[f[x] - g[x], x] == f'[x] - g'[x]
D[f[x] - g[x], {x, 2}] == f''[x] - g''[x]

(* Multiplication by constant *)
D[a * f[x], x] == a * f'[x]
D[a * f[x], {x, 2}] == a * f''[x]

(* Dot product *)
D[f[x] . g[x], x] == f[x] . g'[x] + f'[x] . g[x]
D[f[x] . g[x], {x, 2}] == f[x] . g''[x] + 2 * f'[x] . g'[x] + f''[x] . g[x]

(* Hadamard product *)
D[f[x] * g[x], x] == f[x] * g'[x] + f'[x] * g[x]
D[f[x] * g[x], {x, 2}] == f[x] * g''[x] + 2 * f'[x] * g'[x] + f''[x] * g[x]

(* Inverse *)
D[Inverse[f[x]], x] == -Inverse[f[x]] . f'[x] . Inverse[f[x]]
D[Inverse[f[x]], {x, 2}] == Inverse[f[x]] . (2 * f'[x] . Inverse[f[x]] . f'[x] - f''[x]) . Inverse[f[x]]

(* Transpose *)
D[Transpose[f[x]], x] == Transpose[f'[x]]
D[Transpose[f[x]], {x, 2}] == Transpose[f''[x]]

(* Trace *)
D[Tr[f[x]], x] == Tr[f'[x]]
D[Tr[f[x]], {x, 2}] == Tr[f''[x]]
D[Tr[MatrixPower[f[x], n]], x] == n * Tr[MatrixPower[f[x], n - 1] . f'[x]]
D[Tr[MatrixExp[f[x]]], x] == Tr[MatrixExp[f[x]] . f'[x]]
D[Tr[MatrixLog[f[x]]], x] == Tr[Inverse[f[x]] . f'[x]]
D[Tr[MatrixFunction[p, f[x]]], x] == Tr[MatrixFunction[p', f[x]] . f'[x]]

(* Determinant *)
D[Det[f[x]], x] == Det[f[x]] * Tr[Inverse[f[x]] . f'[x]]
D[Det[f[x]], {x, 2}] == Det[f[x]] * (Tr[Inverse[f[x]] . f'[x]]^2 + Tr[Inverse[f[x]] . (f''[x] - f'[x] . Inverse[f[x]] . f'[x])])
D[Det[MatrixPower[f[x], n]], x] == n * Det[MatrixPower[f[x], n]] * Tr[Inverse[f[x]] . f'[x]]
D[Det[MatrixExp[f[x]]], x] == Det[MatrixExp[f[x]]] * Tr[f'[x]]
D[Det[MatrixLog[f[x]]], x] == Det[MatrixLog[f[x]]] * Tr[f'[x] . Inverse[f[x]] . Inverse[MatrixLog[f[x]]]]
D[Det[MatrixFunction[p, f[x]]], x] == Det[MatrixFunction[p, f[x]]] * Tr[f'[x] . MatrixFunction[p', f[x]] . Inverse[MatrixFunction[p, f[x]]]]

(* Logarithm of determinant *)
D[Log[Det[f[x]]], x] == Tr[Inverse[f[x]] . f'[x]]
D[Log[Det[f[x]]], {x, 2}] == Tr[Inverse[f[x]] . (f''[x] - f'[x] . Inverse[f[x]] . f'[x])]
D[Log[Det[MatrixPower[f[x], n]]], x] == n * Tr[Inverse[f[x]] . f'[x]]
D[Log[Det[MatrixExp[f[x]]]], x] == Tr[f'[x]]
D[Log[Det[MatrixLog[f[x]]]], x] == Tr[f'[x] . Inverse[f[x]] . Inverse[MatrixLog[f[x]]]]
D[Log[Det[MatrixFunction[p, f[x]]]], x] == Tr[f'[x] . MatrixFunction[p', f[x]] . Inverse[MatrixFunction[p, f[x]]]]

Last edited by lanxiyu (2024-12-11 22:49:16)