Adjugate
Adjugate
Adjugate[m]
给出方阵 m 的伴随矩阵.
更多信息
- 伴随矩阵也称为伴随阵，其英文称为 adjugate matrix、classical adjoint matrix 或 adjunct matrix.
- 可逆矩阵 m 的伴随矩阵由 Inverse[m]Det[m] 给出.
- 矩阵 m 与其伴随矩阵的矩阵乘积等于 m 的行列式乘以与 m 相同大小的单位矩阵.
- 矩阵 m 可以是数字或符号，但必须是方阵.
- Adjugate[m] 的 StandardForm 和 TraditionalForm 格式为
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范例
打开所有单元 关闭所有单元基本范例 (3)
Adjugate[{{5, 4}, {4, 11}}]m = {{a, b}, {c, d}};Adjugate[m]//MatrixFormm = {{15, 3, 4}, {3, 15, 11}, {-2, 1, 7}};Adjugate[m].m == Det[m]IdentityMatrix[Length[m]]范围 (10)
基本用法 (6)
Adjugate[{{43.2, 25.1}, {27.1, 41.2}}]Adjugate[{{E^5, 1, 3 - 2I}, {1 + I, (π/2), 5}, {0, 1, -4}}]//MatrixFormAdjugate[{{4, 3}, {3, 4}}]N[{{π, Sqrt[2], E}, {(1/Sqrt[2]), 1 + Sqrt[2], 2 π}, {-(1/Sqrt[2]) + 2 π, -1 + Sqrt[2], 2 E - 2 π}}, 12]//MatrixFormAdjugate[%]//MatrixFormAdjugate[{{a, b, c}, {d, e, f}, {g, h, i}}]//MatrixFormAdjugate[m]特殊矩阵 (4)
SparseArray[{{1, 3} -> 1, {2, 2} -> 2, {3, 1} -> 3}, {3, 3}]Adjugate[%]SymmetrizedArray[{{1, 1} -> 3, {2, 2} -> 1, {3, 1} -> -5}, {3, 3}, Symmetric[All]]Adjugate[%]QuantityArray[Partition[Range[9], 3], {"Meters", "Seconds", "Kilograms"}]Adjugate[%]Adjugate[IdentityMatrix[3]]Adjugate[HilbertMatrix[5]]应用 (4)
使用 Adjugate 计算共同因子：
cofactor[m_, {i_, j_}] := Adjugate[m][[j, i]]m = (| | | | | |
| - | - | - | - | - |
| 6 | 0 | 4 | 9 | 5 |
| 1 | 9 | 3 | 1 | 2 |
| 5 | 4 | 5 | 3 | 8 |
| 3 | 9 | 8 | 2 | 5 |
| 4 | 1 | 6 | 6 | 4 |);cofactor[m, {1, 3}]Apply[And, Table[cofactor[m, {i, j}] == (-1) ^ (i + j)Det[Drop[m, {i}, {j}]], {i, Length[m]}, {j, Length[m]}], {0, 1}]使用 Adjugate 计算矩阵的逆：
m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};1 / Det[m] Adjugate[m]//MatrixForm与 Inverse 比较：
Inverse[m]//MatrixForm使用 Adjugate 求解线性方程：
m = {{1, 1, 1}, {1, 2, 3}, {1, 4, 9}};
b = {1, 2, 3};Adjugate[m].b / Det[m]与 LinearSolve 比较：
LinearSolve[m, b]定义一个函数用于计算以隐式笛卡尔坐标方程表示的曲面的高斯曲率：
gaussianCurvature[f_, {x_, y_, z_}] := Module[{grad, hess},
grad = Grad[f, {x, y, z}];
hess = Grad[grad, {x, y, z}];
grad.Adjugate[hess].grad / (grad.grad) ^ 2]icosurf[x_, y_, z_] := 1 - x^2 - y^2 - 2 x (x^4 - 10 x^2 y^2 + 5 y^4) z - z^2 + 5 (x^2 + y^2)^2 z^2 - 5 (x^2 + y^2) z^4 + z^6 - (x^2 + y^2 + z^2)^3icogc[x_, y_, z_] = Simplify[gaussianCurvature[icosurf[x, y, z], {x, y, z}]];Short[%, 10]ContourPlot3D[icosurf[x, y, z] == 0, {x, -9 / 8, 9 / 8}, {y, -9 / 8, 9 / 8}, {z, -9 / 8, 9 / 8}, ColorFunction -> Function[{x, y, z}, ColorData["ThermometerColors"][LogisticSigmoid[10icogc[x, y, z]]]], ColorFunctionScaling -> False, Mesh -> None, PlotPoints -> 31]属性和关系 (5)
m.Adjugate[m] 等于 Det[m] 乘以与其尺寸相同的单位矩阵：
m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};m.Adjugate[m] == Det[m]IdentityMatrix[Length[m]]Inverse[m] 等于伴随矩阵除以行列式：
m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};Inverse[m] == Adjugate[m] / Det[m]对于 n×n 矩阵 m，Adjugate[m] 等于 LinearSolve[m,Det[m]IdentityMatrix[n]]：
m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};Adjugate[m] == LinearSolve[m, Det[m]IdentityMatrix[Length[m]]]对于 n×n 矩阵 m，Det[Adjugate[m]]==Det[m]n-1：
m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};Det[Adjugate[m]] == Det[m]^Length[m] - 1m = RandomInteger[9, {5, 5}];
Reverse[Minors[m], {1, 2}] == Transpose[Adjugate[m]]Table[(-1) ^ (i + j), {i, Length[m]}, {j, Length[m]}]
巧妙范例 (2)
adjugatePolynomial[A_ ? SquareMatrixQ, x_] := With[{n = Length[A]}, Sum[Sum[Product[((-1)^id[[l]] + 1/l^id[[l]](id[[l]]!))Tr[MatrixPower[A, l]]^id[[l]], {l, n - 1}], {id, FrobeniusSolve[Range[n - 1], n - s - 1]}]x^s, {s, 0, n - 1}]]m = {{1, 1, 1}, {1, 2, 3}, {1, 4, 9}};ap[x_] = adjugatePolynomial[m, x]MatrixFunction[ap, m]//FullSimplifyAdjugate[m]iteratedAdjugate[A_ ? SquareMatrixQ, k_Integer ? NonNegative] := With[{n = Length[A]},
Det[A]^((n - 1)^k - (-1)^k/n) - Boole[OddQ[k]]If[OddQ[k], Adjugate[A], A]]m = {{1, 3, 5}, {7, 1, -1}, {8, 4, 1}};Table[iteratedAdjugate[m, k], {k, 0, 5}]NestList[Adjugate, m, 5]
Wolfram Research (2021)，Adjugate，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Adjugate.html.
文本
Wolfram Research (2021)，Adjugate，Wolfram 语言函数，https://reference.wolfram.com/language/ref/Adjugate.html.
CMS
Wolfram 语言. 2021. "Adjugate." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/Adjugate.html.
APA
Wolfram 语言. (2021). Adjugate. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/Adjugate.html 年