矩阵分解----Cholesky分解

Posted 2021-01-20 hurenkebi1

矩阵分解是将矩阵拆解成多个矩阵的乘积，常见的分解方法有 三角分解法、QR分解法、奇异值分解法。三角分解法是将原方阵分解成一个上三角矩阵和一个下三角矩阵，这种分解方法叫做LU分解法。进一步，如果待分解的矩阵A是正定的，则A可以唯一的分解为

[{f{A = L}}{{f{L}}^{f{T}}}]

其中L是下三角矩阵。下面以三维矩阵进行简单说明：

[egin{array}{ccccc}
{f{A = L}}{{f{L}}^{f{T}}}{ m{ = }} & left[ {egin{array}{*{20}{c}}
{{L_{11}}}&0&0\
{{L_{21}}}&{{L_{22}}}&0\
{{L_{31}}}&{{L_{32}}}&{{L_{33}}}
end{array}} ight]left[ {egin{array}{*{20}{c}}
{{L_{11}}}&{{L_{21}}}&{{L_{31}}}\
0&{{L_{22}}}&{{L_{32}}}\
0&0&{{L_{33}}}
end{array}} ight]\
= & left[ {egin{array}{*{20}{c}}
{L_{11}^2}&{}&{left( {symmetric} ight)}\
{{L_{21}}{L_{11}}}&{L_{21}^2 + L_{22}^2}&{}\
{{L_{31}}{L_{11}}}&{{L_{31}}{L_{21}} + {L_{32}}{L_{22}}}&{L_{31}^2 + L_{32}^2 + L_{33}^2}
end{array}} ight]\
= & left[ {egin{array}{*{20}{c}}
{{A_{11}}}&{{A_{12}}}&{{A_{13}}}\
{{A_{21}}}&{{A_{22}}}&{{A_{23}}}\
{{A_{31}}}&{{A_{32}}}&{{A_{33}}}
end{array}} ight]
end{array}]

由上式可以得到

[{f{L}} = left[ {egin{array}{*{20}{c}}
{sqrt {{A_{11}}} }&0&0\
{frac{{{A_{21}}}}{{{L_{11}}}}}&{sqrt {{A_{22}} - L_{21}^2} }&0\
{frac{{{A_{31}}}}{{{L_{11}}}}}&{frac{{{A_{32}} - {L_{31}}{L_{21}}}}{{{L_{22}}}}}&{sqrt {{A_{33}} - L_{31}^2 - L_{32}^2} }
end{array}} ight]]

进一步进行多维扩展得到实数矩阵分解的表达式为

[egin{array}{l}
{L_{j,j}} = sqrt {{A_{j,j}} - sumlimits_{k = 1}^{j - 1} {L_{j,k}^2} } \
{L_{i,j}} = frac{{left( {{A_{i,j}} - sumlimits_{k = 1}^{j - 1} {{L_{i,k}}{L_{j,k}}} } ight)}}{{{L_{j,j}}}}for,i > j
end{array}]

对于复数矩阵可以得到类似的公式

[egin{array}{l}
{L_{j,j}} = sqrt {{A_{j,j}} - sumlimits_{k = 1}^{j - 1} {{L_{j,k}}L_{j,k}^ * } } \
{L_{i,j}} = frac{{left( {{A_{i,j}} - sumlimits_{k = 1}^{j - 1} {{L_{i,k}}L_{j,k}^ * } } ight)}}{{{L_{j,j}}}}for,i > j
end{array}]

上式过程叫做Cholesky分解，由公式可知，该方法存在开根号的操作，在硬件实现中复杂度较高。一般采用LDL分解法来规避这个问题。