Matrix Factorization through LU / PLU / LUP Decomposition using Elementary Row/Column Operations

1. Given any \(M \times N\) Matrix \(A\) as follows
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\)   ...(1)
   
   LU / PLU / LUP Decomposition of Matrix \(A\) invloves Factoring Matrix \(A\) into any one of the following
   1. A \(M \times M\) Lower Uni-Triangular Square Matrix \(L\) and a \(M\times N\) Upper Triangular/Trapezoidal Matrix \(U\). Depending on the Matrix \(A\), the \(LU\) Matrix May have to be Optionally Pre-Multiplied with of an \(M \times M\) Permutation Matrix \(P\), thus making it a PLU Decomposition. Following gives the Layout of Matrices for such Decomposition
      
      \(A=(P)LU=(P)\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & l_{32} & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & u_{22} & u_{23} & ... & u_{2n} \\ 0 & 0 & u_{33} & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & u_{mn}\end{bmatrix}\)   ...(2)
      
      The Matrix \(P\) has been given in Paranthesis to indicate that it May or May not be Present depending on the Matrix \(A\).
   2. A \(M \times N\) Lower Triangular/Trapezoidal Matrix \(L\) and a \(N\times N\) Upper Uni-Triangular Square Matrix \(U\). Depending on the Matrix \(A\), the \(LU\) Matrix May have to be Optionally Post-Multiplied with an \(N \times N\) Permutation Matrix \(P\), thus making it a LUP Decomposition. Following gives the Layout of Matrices for such Decomposition
      
      \(A=LU(P)=(P)\begin{bmatrix}l_{11} & 0 & 0 & \cdots & 0_{1n} \\l_{21} & l_{22} & 0 & \cdots & 0_{2n} \\l_{31} & l_{32} & l_{33} & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & l_{mn}\end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} & ... & u_{1n}\\ 0 & 1 & u_{23} & ... & u_{2n} \\ 0 & 0 & 1 & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & 1_{nn}\end{bmatrix}(P)\)   ...(3)
      
      The Matrix \(P\) has been given in Paranthesis to indicate that it May or May not be Present depending on the Matrix \(A\).
2. Following gives the Algorithm for (P)LU Decomposition of Matrix \(A\) as given in equation (1) above
   1. The Matrix \(A\) in equation (1) can be given as Matrix \(A\) Pre-Multiplied by \(M\times M\) Identity Matrix as follows
      
      \(A=IA=LU=\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\0 & 1 & 0 & \cdots & 0_{2m} \\0 & 0 & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\)   (Initially Matrix \(L=I\) and Matrix \(U=A\)) ...(4)
      
      
   2. If the Element \(a_{11}=0\) in Matrix \(U\) / Matrix \(A\) , then Interchange \(1^{st}\) Row of the Matrix \(U\) with the first \(k^{th}\) Row (where \(k > 1\) ) for which \(a_{k1}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=LU=ILIU=P_{1k}P_{1k}LP_{1k}P_{1k}U\hspace{.6cm}\Rightarrow A=P_{1k}LU\)   (\(\because P_{1k}P_{1k}=I\)) ...(5)
      
      where \(P_{1k}\) is the Permutation Matrix that Interchanges the Elements of the Row \(1\) and Row \(k\) when Pre-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Post-Multiplied with Matrix \(L\) it Interchanges the Elements of the Column \(1\) and Column \(k\) of Matrix \(L\).
      
      Please note that this step is Not Required when \(a_{11}\neq 0\) in Matrix \(U\) / Matrix \(A\). Under such conditions Permutation Matrix \(P_{1k}\) can be assumed to be same as Identity Matrix \(I\).
   3. If the Element \(a_{11}\neq 0\), then For every Row \(k > 1\), perform the following Row Operation on Matrix \(U\)
      
      \(R_k = R_k -\frac{a_{k1}}{a_{11}} \times R_1\)   ...(6)
      
      Simultaneously, if the Element \(a_{11}\neq 0\), then for every Row \(k > 1\), perform the following Row Operation on Matrix \(L\)
      
      \(R_k=R_k+\frac{a_{k1}}{a_{11}} \times R_1\)   ...(7)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=P_{1k}LU=P_{1k}\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & 0 & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & 0 & 0 & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & a_{22} & a_{23} & ... & a_{2n} \\ 0 & a_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix}\)   ...(8)
      
      
   4. If the Element \(a_{22}=0\) in Matrix \(U\), then Interchange \(2^{nd}\) Row of the Matrix \(U\) with the first \(k^{th}\) Row (where \(k > 2\) ) for which \(a_{k2}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=P_{1k}LU=P_{1k}ILIU=P_{1k}P_{2k}P_{2k}LP_{2k}P_{2k}U\hspace{.6cm}\Rightarrow A=P_{1k}P_{2k}LU\)   (\(\because P_{2k}P_{2k}=I\)) ...(9)
      
      where \(P_{2k}\) is the Permutation Matrix that Interchanges the Elements of the Row \(2\) and Row \(k\) when Pre-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Post-Multiplied with Matrix \(L\) it Interchanges the Elements of the Column \(2\) and Column \(k\) of Matrix \(L\).
      
      Please note that this step is Not Required when \(a_{22}\neq 0\) in Matrix \(U\). Under such conditions Permutation Matrix \(P_{2k}\) can be assumed to be same as Identity Matrix \(I\).
   5. If the Element \(a_{22}\neq 0\), then For every Row \(k > 2\), perform the following Row Operation on Matrix \(U\)
      
      \(R_k = R_k -\frac{a_{k2}}{a_{22}} \times R_2\)   ...(10)
      
      Simultaneously, if the Element \(a_{22}\neq 0\), then for every Row \(k > 2\), perform the following Row Operation on Matrix \(L\) for all Columns \(> 1\)
      
      \(R_k=R_k+\frac{a_{k2}}{a_{22}} \times R_2\)   ...(11)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=P_{1k}P_{2k}LU=P_{1k}P_{2k}\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & l_{32} & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & 0 & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & u_{22} & u_{23} & ... & u_{2n} \\ 0 & 0 & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & a_{m3} & ... & a_{mn}\end{bmatrix}\)   ...(12)
      
      
   6. If the Element \(a_{33}=0\) in Matrix \(U\), then Interchange \(3^{rd}\) Row of the Matrix \(U\) with the first \(k^{th}\) Row (where \(k > 3\) ) for which \(a_{k3}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=P_{1k}P_{2k}LU=P_{1k}P_{2k}ILIU=P_{1k}P_{2k}P_{3k}P_{3k}LP_{3k}P_{3k}U\hspace{.6cm}\Rightarrow A=P_{1k}P_{2k}P_{3k}LU\)   (\(\because P_{3k}P_{3k}=I\)) ...(13)
      
      where \(P_{3k}\) is the Permutation Matrix that Interchanges the Elements of the Row \(3\) and Row \(k\) when Pre-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Post-Multiplied with Matrix \(L\) it Interchanges the Elements of the Column \(3\) and Column \(k\) of Matrix \(L\).
      
      Please note that this step is Not Required when \(a_{33}\neq 0\) in Matrix \(U\). Under such conditions Permutation Matrix \(P_{3k}\) can be assumed to be same as Identity Matrix \(I\).
   7. If the Element \(a_{33}\neq 0\), then For every Row \(k > 3\), perform the following Row Operation on Matrix \(U\)
      
      \(R_k = R_k -\frac{a_{k3}}{a_{33}} \times R_3\)   ...(14)
      
      Simultaneously, if the Element \(a_{33}\neq 0\), then for every Row \(k > 3\), perform the following Row Operation on Matrix \(L\) for all Columns \(> 2\)
      
      \(R_k=R_k+\frac{a_{k3}}{a_{33}} \times R_3\)   ...(15)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=P_{1k}P_{2k}P_{3k}LU=P_{1k}P_{2k}P_{3k}\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & l_{32} & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & u_{22} & u_{23} & ... & u_{2n} \\ 0 & 0 & u_{33} & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & a_{mn}\end{bmatrix}\)   ...(16)
      
      
   8. Once the Above Mentioned Steps are Performed and Completed for all Rows \(< M\), the Matrices \(L\) and \(U\) shall get converted to the desired Lower Uni-Triangular Square Matrix and the Upper Triangular/Trapezoidal Matrix respectively. Also the Matrix Product of Permutation Matrices \(P_{1k}P_{2k}P_{3k}...P_{m-1k}\) can be represented as a Single Permutation Matrix \(P\), thus giving the following Factorization
      
      \(A=PLU=P\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & l_{32} & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & u_{22} & u_{23} & ... & u_{2n} \\ 0 & 0 & u_{33} & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & u_{mn}\end{bmatrix}\)   ...(17)
      
      In case No Interchages between the Rows are required, then the Permutation Matrix \(P\) is Same as the Identity Matrix \(I\). And hence,
      
      \(A=ILU=LU=\begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1m} \\l_{21} & 1 & 0 & \cdots & 0_{2m} \\l_{31} & l_{32} & 1 & \cdots & 0_{3m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & 1_{mm}\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} & ... & u_{1n}\\ 0 & u_{22} & u_{23} & ... & u_{2n} \\ 0 & 0 & u_{33} & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & u_{mn}\end{bmatrix}\)   ...(18)
      
      
3. Following gives the Algorithm for LU(P) Decomposition of a Matrix
   1. The Matrix \(A\) in equation (1) can be given as Matrix \(A\) Post-Multiplied by \(N\times N\) Identity Matrix as follows
      
      \(A=AI=LU=\begin{bmatrix} a_{11} & a_{12} & a_{13} & ... & a_{1n}\\ a_{21} & a_{22} & a_{23} & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix} \begin{bmatrix}1 & 0 & 0 & \cdots & 0_{1n} \\0 & 1 & 0 & \cdots & 0_{2n} \\0 & 0 & 1 & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1_{nn}\end{bmatrix}\)   (Initially Matrix \(L=A\) and Matrix \(U=I\)) ...(19)
      
      
   2. If the Element \(a_{11}=0\) in Matrix \(L\) / Matrix \(A\) , then Interchange \(1^{st}\) Column of the Matrix \(L\) with the first \(k^{th}\) Column (where \(k > 1\) ) for which \(a_{1k}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=LU=LIUI=LP_{1k}P_{1k}UP_{1k}P_{1k}\hspace{.6cm}\Rightarrow A=LUP_{1k}\)   (\(\because P_{1k}P_{1k}=I\)) ...(20)
      
      where \(P_{1k}\) is the Permutation Matrix that Interchanges the Elements of the Column \(1\) and Column \(k\) when Post-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Pre-Multiplied with Matrix \(U\) it Interchanges the Elements of the Row \(1\) and Row \(k\) of Matrix \(U\).
      
      Please note that this step is Not Required when \(a_{11}\neq 0\) in Matrix \(L\) / Matrix \(A\). Under such conditions Permutation Matrix \(P_{1k}\) can be assumed to be same as Identity Matrix \(I\).
   3. If the Element \(a_{11}\neq 0\), then For every Column \(k > 1\), perform the following Column Operation on Matrix \(L\)
      
      \(C_k = C_k -\frac{a_{1k}}{a_{11}} \times C_1\)   ...(21)
      
      Simultaneously, if the Element \(a_{11}\neq 0\), then for every Column \(k > 1\), perform the following Row Operation on Matrix \(U\)
      
      \(C_k=C_k+\frac{a_{1k}}{a_{11}} \times C_1\)   ...(22)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=LUP_{1k}=\begin{bmatrix} l_{11} & 0 & 0 & ... & 0\\ l_{21} & a_{22} & a_{23} & ... & a_{2n} \\ l_{31} & a_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & a_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix} \begin{bmatrix}1 & u_{11} & u_{12} & \cdots & u_{1n} \\0 & 1 & 0 & \cdots & 0_{2n} \\0 & 0 & 1 & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1_{nn}\end{bmatrix}P_{1k}\)   ...(23)
      
      
   4. If the Element \(a_{22}=0\) in Matrix \(L\), then Interchange \(2^{nd}\) Column of the Matrix \(L\) with the first \(k^{th}\) Column (where \(k > 2\) ) for which \(a_{2k}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=LUP_{1k}=LIUIP_{1k}=LP_{2k}P_{2k}UP_{2k}P_{2k}P_{1k}\hspace{.6cm}\Rightarrow A=LUP_{2k}P_{1k}\)   (\(\because P_{2k}P_{2k}=I\)) ...(24)
      
      where \(P_{2k}\) is the Permutation Matrix that Interchanges the Elements of the Column \(2\) and Column \(k\) when Post-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Pre-Multiplied with Matrix \(U\) it Interchanges the Elements of the Row \(2\) and Row \(k\) of Matrix \(U\).
      
      Please note that this step is Not Required when \(a_{22}\neq 0\) in Matrix \(L\). Under such conditions Permutation Matrix \(P_{2k}\) can be assumed to be same as Identity Matrix \(I\).
   5. If the Element \(a_{22}\neq 0\), then For every Column \(k > 2\), perform the following Row Operation on Matrix \(L\)
      
      \(C_k = C_k -\frac{a_{2k}}{a_{22}} \times C_2\)   ...(25)
      
      Simultaneously, if the Element \(a_{22}\neq 0\), then for every Column \(k > 2\), perform the following Row Operation on Matrix \(U\) for all Rows \(> 1\)
      
      \(C_k=C_k+\frac{a_{2k}}{a_{22}} \times C_2\)   ...(26)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=LUP_{2k}P_{1k}=\begin{bmatrix} l_{11} & 0 & 0 & ... & 0\\ l_{21} & l_{22} & 0 & ... & 0 \\ l_{31} & l_{32} & a_{33} & ... & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & a_{m3} & ... & a_{mn}\end{bmatrix} \begin{bmatrix}1 & u_{11} & u_{12} & \cdots & u_{1n} \\0 & 1 & u_{23} & \cdots & u_{2n} \\0 & 0 & 1 & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1_{nn}\end{bmatrix}P_{2k}P_{1k}\)   ...(27)
      
      
   6. If the Element \(a_{33}=0\) in Matrix \(L\), then Interchange \(3^{rd}\) Column of the Matrix \(L\) with the first \(k^{th}\) Column (where \(k > 3\) ) for which \(a_{3k}\neq 0\) by performing Matrix Multiplications as follows
      
      \(A=LUP_{2k}P_{1k}=LIUIP_{2k}P_{1k}=LP_{3k}P_{3k}UP_{3k}P_{3k}P_{2k}P_{1k}\hspace{.6cm}\Rightarrow A=LUP_{3k}P_{2k}P_{1k}\)   (\(\because P_{3k}P_{3k}=I\)) ...(28)
      
      where \(P_{3k}\) is the Permutation Matrix that Interchanges the Elements of the Column \(3\) and Columb \(k\) when Post-Multiplied with Matrix \(L\) and Matrix \(U\). Also when Pre-Multiplied with Matrix \(U\) it Interchanges the Elements of the Row \(3\) and Row \(k\) of Matrix \(U\).
      
      Please note that this step is Not Required when \(a_{33}\neq 0\) in Matrix \(L\). Under such conditions Permutation Matrix \(P_{3k}\) can be assumed to be same as Identity Matrix \(I\).
   7. If the Element \(a_{33}\neq 0\), then For every Column \(k > 3\), perform the following Column Operation on Matrix \(L\)
      
      \(C_k = C_k -\frac{a_{3k}}{a_{33}} \times C_3\)   ...(29)
      
      Simultaneously, if the Element \(a_{33}\neq 0\), then for every Column \(k > 3\), perform the following Columb Operation on Matrix \(L\) for all Rows \(> 2\)
      
      \(C_k=C_k+\frac{a_{3k}}{a_{33}} \times C_3\)   ...(30)
      
      Once the above Row Operations have been performed on Matrices \(L\) and \(U\), the Factors of Matrix \(A\) shall appear as follows
      
      \(A=LUP_{3k}P_{2k}P_{1k}=\begin{bmatrix} l_{11} & 0 & 0 & ... & 0\\ l_{21} & l_{22} & 0 & ... & 0 \\ l_{31} & l_{32} & l_{33} & ... & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & ... & a_{mn}\end{bmatrix} \begin{bmatrix}1 & u_{11} & u_{12} & \cdots & u_{1n} \\0 & 1 & u_{23} & \cdots & u_{2n} \\0 & 0 & 1 & \cdots & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \cdots & 1_{nn}\end{bmatrix}P_{3k}P_{2k}P_{1k}\)   ...(31)
      
      
   8. Once the Above Mentioned Steps are Performed and Completed for all Columns \(< N\), the Matrices \(L\) and \(U\) shall get converted to the desired Lower Triangular/Trapezoidal Matrix and the Upper Uni-Triangular Square Matrix respectively. Also the Matrix Product of Permutation Matrices \(P_{m-1k}...P_{3k}P_{2k}P_{1k}\) can be represented as a Single Permutation Matrix \(P\), thus giving the following Factorization
      
      \(A=LUP=\begin{bmatrix}l_{11} & 0 & 0 & \cdots & 0_{1n} \\l_{21} & l_{22} & 0 & \cdots & 0_{2n} \\l_{31} & l_{32} & l_{33} & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & l_{mn}\end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} & ... & u_{1n}\\ 0 & 1 & u_{23} & ... & u_{2n} \\ 0 & 0 & 1 & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & 1_{nn}\end{bmatrix}P\)   ...(32)
      
      In case No Interchages between the Columns are required, then the Permutation Matrix \(P\) is Same as the Identity Matrix \(I\). And hence,
      
      \(A=LUI=LU=\begin{bmatrix}l_{11} & 0 & 0 & \cdots & 0_{1n} \\l_{21} & l_{22} & 0 & \cdots & 0_{2n} \\l_{31} & l_{32} & l_{33} & \cdots & 0_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\l_{m1} & l_{m2} & l_{m3} & \cdots & l_{mn}\end{bmatrix} \begin{bmatrix} 1 & u_{12} & u_{13} & ... & u_{1n}\\ 0 & 1 & u_{23} & ... & u_{2n} \\ 0 & 0 & 1 & ... & u_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & ... & 1_{nn}\end{bmatrix}\)   ...(33)