Mark the following either true or false. Justification is required. (a) In finding an LU factorization, only row replacement is al- lowed, where R, is replaced by R, + kR, for i > j. (b) Every matrix has an LU factorization.
Mark the following either true or false. Justification is required. (a) In finding an LU factorization, only row replacement is al- lowed, where R, is replaced by R, + kR, for i > j. (b) Every matrix has an LU factorization.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**True or False Statements: LU Factorization**
**Instructions:** Determine whether the following statements are true or false. Justification is required for each answer.
**(a)** In finding an LU factorization, only row replacement is allowed, where \( R_i \) is replaced by \( R_i + kR_j \) for \( i > j \).
**(b)** Every matrix has an LU factorization.
Expert Solution

Step 1: Justification of part a
False,
Row replacement is not the only operation allowed in LU factorization. LU factorization, specifically using Gaussian elimination, involves three types of elementary row operations:
- Row interchange: Interchanging the positions of two rows.
- Row scaling: Multiplying a row by a nonzero constant.
- Row addition: Adding a multiple of one row to another row.
These operations are used to transform the original matrix into an upper triangular form while simultaneously constructing a lower triangular matrix
that keeps track of the row operations performed. The row operations are chosen in a way that the resulting matrix
is upper triangular, and the matrix
is lower triangular with ones on the diagonal.
So, row scaling (multiplying rows by constants) and row addition (adding multiples of rows) are indeed allowed operations in LU decomposition.
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