Understand various matrix operations, matrix decompositions, factorization and related operations
LU Decomposition
LU decomposition breaks a matrix into a product of a lower and an upper triangular matrix, simplifying linear equation solving and matrix inversion.
LU Decomposition is the process of factoring a matrix \( A \) into two triangular matrices: a lower triangular matrix \( L \) and an upper triangular matrix \( U \), such that:
\( A = L \cdot U \)
Matrix A
\( A = \begin{bmatrix}
2 & 4 & 3 & 5 \\
-4 & -7 & -5 & -8 \\
6 & 8 & 2 & 9 \\
4 & 9 & -2 & 14
\end{bmatrix} \)
Step 1: First Column Elimination
We eliminate the entries below the first pivot (2 in row 1):
- \( R_2 \leftarrow R_2 + 2 \cdot R_1 \) → Multiplier: -2
- \( R_3 \leftarrow R_3 - 3 \cdot R_1 \) → Multiplier: 3
- \( R_4 \leftarrow R_4 - 2 \cdot R_1 \) → Multiplier: 2
Updated matrix U:
\( U_1 = \begin{bmatrix}
2 & 4 & 3 & 5 \\
0 & 1 & 1 & 2 \\
0 & -4 & -7 & -6 \\
0 & 1 & -8 & 4
\end{bmatrix} \)
Partial L matrix:
\( L = \begin{bmatrix}
1 & 0 & 0 & 0 \\
-2 & 1 & 0 & 0 \\
3 & 0 & 1 & 0 \\
2 & 0 & 0 & 1
\end{bmatrix} \)
Step 2: Second Column Elimination
- \( R_3 \leftarrow R_3 + 4 \cdot R_2 \) → Multiplier: -4
- \( R_4 \leftarrow R_4 - 1 \cdot R_2 \) → Multiplier: 1
Updated matrix U:
\( U_2 = \begin{bmatrix}
2 & 4 & 3 & 5 \\
0 & 1 & 1 & 2 \\
0 & 0 & -3 & 2 \\
0 & 0 & -9 & 2
\end{bmatrix} \)
Updated L matrix:
\( L = \begin{bmatrix}
1 & 0 & 0 & 0 \\
-2 & 1 & 0 & 0 \\
3 & -4 & 1 & 0 \\
2 & 1 & 0 & 1
\end{bmatrix} \)
Step 3: Third Column Elimination
- \( R_4 \leftarrow R_4 - 3 \cdot R_3 \) → Multiplier: 3
Final matrix U:
\( U = \begin{bmatrix}
2 & 4 & 3 & 5 \\
0 & 1 & 1 & 2 \\
0 & 0 & -3 & 2 \\
0 & 0 & 0 & -4
\end{bmatrix} \)
Final L matrix:
\( L = \begin{bmatrix}
1 & 0 & 0 & 0 \\
-2 & 1 & 0 & 0 \\
3 & -4 & 1 & 0 \\
2 & 1 & 3 & 1
\end{bmatrix} \)