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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<math>\Rightarrow \left\{
 
<math>\Rightarrow \left\{
 
   \begin{array}{lr}
 
   \begin{array}{lr}
     u_1 = x_1 & \\
+
     u_1 = m_1 & \\
     lu_1 = x_2 & \\
+
     lu_1 = m_2 & \\
     u_3 = x_3& \\
+
     u_3 = m_3& \\
     lu_3 + u_2 = x_4
+
     lu_3 + u_2 = m_4
 
   \end{array}
 
   \end{array}
 
\right.
 
\right.
 
</math>
 
</math>
  
Let <math>x_1 = x_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.
+
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.