LU decomposition
  • A procedure for decomposing an  matrix  into a product of a lower triangular matrix  L and an upper triangular matrix U,  LU=A
  • LU decomposition is implemented in the Wolfram  language as LUDecomposition.
  • Written explicitly for a  matrix, the decomposition is

  • This gives three types of equations

Example:

In the LU decomposition of the matrix

\begin{vmatrix} 2 &2 \\ 4 & 9 \end{vmatrix} , if the diagonal elements of U are both 1, then the lower diagonal entry l22l of L is (GATE CS 2015)
(A) 4
(B) 5
(C) 6
(D) 7

Soluation:

LU decomposition factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.

\begin{vmatrix} L11 & 0 \\ L21 & L22 \end{vmatrix}*\begin{vmatrix} 1 & U12 \\ 0 & 1 \end{vmatrix}

 

L21 * U12 + L22 * 1 = 9 ------ (1)

We need to find L21 and U12

L21 *  1 + L22 * 0  = 4

L21 = 4

L11 * U12 + 0 * 1 = 2

U11 = 2

U12 = 1

Putting value of L21 and U12 in (1),

we get 4 * 1 + L22 * 1 = 9

L22 = 5

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