Difference between revisions of "TDSM 8.9"
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(Created page with "LU factorization of a matrix is not necessarily unique. Example: proof for <math>2 \times 2</math> square matrix: Let <math> L = [10l1] ...") |
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Line 30: | Line 30: | ||
<math>\Rightarrow \left\{ | <math>\Rightarrow \left\{ | ||
\begin{array}{lr} | \begin{array}{lr} | ||
− | u_1 = | + | u_1 = m_1 & \\ |
− | lu_1 = | + | lu_1 = m_2 & \\ |
− | u_3 = | + | u_3 = m_3& \\ |
− | lu_3 + u_2 = | + | lu_3 + u_2 = m_4 |
\end{array} | \end{array} | ||
\right. | \right. | ||
</math> | </math> | ||
− | Let <math> | + | Let <math>m_1 = m_2 = 0 \Rightarrow</math> There are 3 equations for 4 variables <math>\Rightarrow</math> There are many value for <math>l</math> satisfies the equations. |
<math>\Rightarrow</math> LU factorization of <math>M</math> not unique. | <math>\Rightarrow</math> LU factorization of <math>M</math> not unique. |
Latest revision as of 01:47, 12 December 2017
LU factorization of a matrix is not necessarily unique. Example: proof for 2×2 square matrix:
Let L=[10l1], U=[u1u30u2]
⇒LU=[u1u3lu1lu3+u2]
Let M=[m1m3m2m4]=LU
⇒{u1=m1lu1=m2u3=m3lu3+u2=m4
Let m1=m2=0⇒ There are 3 equations for 4 variables ⇒ There are many value for l satisfies the equations.
⇒ LU factorization of M not unique.