lecture 4: lu decomposition and matrix inversethanghuynh.org/teaching/math102_lecture_4.pdfgaussian...
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Gaussian elimination revisited
ⶠExample. Keeping track of the elementary matrices duringGaussian elimination on ðŽ:
ðŽ = â¡â¢â£2 14 â6
â€â¥âŠ
ðžðŽ = â¡â¢â£
1 0â2 1
â€â¥âŠ
â¡â¢â£2 14 â6
â€â¥âŠ
= â¡â¢â£2 10 â8
â€â¥âŠ
.
Note thatðŽ = ðžâ1 â¡â¢
â£2 10 â8
â€â¥âŠ
= â¡â¢â£1 02 1
â€â¥âŠ
â¡â¢â£2 10 â8
â€â¥âŠ
We factored ðŽ as the product of a lower and upper triangularmatrix! We say that ðŽ has triangular factorization.
1
Gaussian elimination revisited
ⶠExample. Keeping track of the elementary matrices duringGaussian elimination on ðŽ:
ðŽ = â¡â¢â£2 14 â6
â€â¥âŠ
ðžðŽ = â¡â¢â£
1 0â2 1
â€â¥âŠ
â¡â¢â£2 14 â6
â€â¥âŠ
= â¡â¢â£2 10 â8
â€â¥âŠ
.
Note thatðŽ = ðžâ1 â¡â¢
â£2 10 â8
â€â¥âŠ
= â¡â¢â£1 02 1
â€â¥âŠ
â¡â¢â£2 10 â8
â€â¥âŠ
We factored ðŽ as the product of a lower and upper triangularmatrix! We say that ðŽ has triangular factorization.
1
Gaussian elimination revisited
ⶠExample. Keeping track of the elementary matrices duringGaussian elimination on ðŽ:
ðŽ = â¡â¢â£2 14 â6
â€â¥âŠ
ðžðŽ = â¡â¢â£
1 0â2 1
â€â¥âŠ
â¡â¢â£2 14 â6
â€â¥âŠ
= â¡â¢â£2 10 â8
â€â¥âŠ
.
Note thatðŽ = ðžâ1 â¡â¢
â£2 10 â8
â€â¥âŠ
= â¡â¢â£1 02 1
â€â¥âŠ
â¡â¢â£2 10 â8
â€â¥âŠ
We factored ðŽ as the product of a lower and upper triangularmatrix! We say that ðŽ has triangular factorization.
1
Gaussian elimination revisited
ⶠExample. Keeping track of the elementary matrices duringGaussian elimination on ðŽ:
ðŽ = â¡â¢â£2 14 â6
â€â¥âŠ
ðžðŽ = â¡â¢â£
1 0â2 1
â€â¥âŠ
â¡â¢â£2 14 â6
â€â¥âŠ
= â¡â¢â£2 10 â8
â€â¥âŠ
.
Note thatðŽ = ðžâ1 â¡â¢
â£2 10 â8
â€â¥âŠ
= â¡â¢â£1 02 1
â€â¥âŠ
â¡â¢â£2 10 â8
â€â¥âŠ
We factored ðŽ as the product of a lower and upper triangularmatrix! We say that ðŽ has triangular factorization.
1
Gaussian elimination revisited
ðŽ = ð¿ð is known as the LU decomposition of ðŽ.
ⶠDefinition.lower triangular
â¡â¢â¢â¢â¢â¢â¢â£
â 0 0 0 0â â 0 0 0â â â 0 0â â â â 0â â â â â
â€â¥â¥â¥â¥â¥â¥âŠ
upper triangular
â¡â¢â¢â¢â¢â¢â¢â£
â â â â â0 â â â â0 0 â â â0 0 0 â â0 0 0 0 â
â€â¥â¥â¥â¥â¥â¥âŠ
2
Gaussian elimination revisited
ðŽ = ð¿ð is known as the LU decomposition of ðŽ.
ⶠDefinition.lower triangular
â¡â¢â¢â¢â¢â¢â¢â£
â 0 0 0 0â â 0 0 0â â â 0 0â â â â 0â â â â â
â€â¥â¥â¥â¥â¥â¥âŠ
upper triangular
â¡â¢â¢â¢â¢â¢â¢â£
â â â â â0 â â â â0 0 â â â0 0 0 â â0 0 0 0 â
â€â¥â¥â¥â¥â¥â¥âŠ
2
ð¿ð decompostion
ⶠExample. Factor ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
as ðŽ = ð¿ð.
ⶠSolution.
ðž1ðŽ =â¡â¢â¢â£
1 0 0â2 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
ðž2(ðž1ðŽ) =â¡â¢â¢â£
1 0 00 1 01 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
ðž3ðž2ðž1ðŽ =â¡â¢â¢â£
1 0 00 1 00 1 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 0 1
â€â¥â¥âŠ
= ð
3
ð¿ð decompostion
ⶠExample. Factor ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
as ðŽ = ð¿ð.
ⶠSolution.
ðž1ðŽ =â¡â¢â¢â£
1 0 0â2 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
ðž2(ðž1ðŽ) =â¡â¢â¢â£
1 0 00 1 01 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
ðž3ðž2ðž1ðŽ =â¡â¢â¢â£
1 0 00 1 00 1 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 0 1
â€â¥â¥âŠ
= ð
3
ð¿ð decompostion
ⶠExample. Factor ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
as ðŽ = ð¿ð.
ⶠSolution.
ðž1ðŽ =â¡â¢â¢â£
1 0 0â2 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
ðž2(ðž1ðŽ) =â¡â¢â¢â£
1 0 00 1 01 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
ðž3ðž2ðž1ðŽ =â¡â¢â¢â£
1 0 00 1 00 1 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 0 1
â€â¥â¥âŠ
= ð
3
ð¿ð decompostion
ⶠExample. Factor ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
as ðŽ = ð¿ð.
ⶠSolution.
ðž1ðŽ =â¡â¢â¢â£
1 0 0â2 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
ðž2(ðž1ðŽ) =â¡â¢â¢â£
1 0 00 1 01 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â2
â2 7 2
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
ðž3ðž2ðž1ðŽ =â¡â¢â¢â£
1 0 00 1 00 1 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â20 8 3
â€â¥â¥âŠ
=â¡â¢â¢â£
2 1 10 â8 â20 0 1
â€â¥â¥âŠ
= ð
3
ð¿ð decompostion
ðž3ðž2ðž1ðŽ = ð
â¹ ðŽ = ðžâ11 ðžâ1
2 ðžâ13 ð
The factor ð¿ is given by
ð¿ = ðžâ11 ðžâ1
2 ðžâ13
=â¡â¢â¢â£
1 0 02 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 0
â1 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 00 â1 1
â€â¥â¥âŠ
=â¡â¢â¢â£
1 0 02 1 0
â1 â1 1
â€â¥â¥âŠ
4
ð¿ð decompostion
ðž3ðž2ðž1ðŽ = ð â¹ ðŽ = ðžâ11 ðžâ1
2 ðžâ13 ð
The factor ð¿ is given by
ð¿ = ðžâ11 ðžâ1
2 ðžâ13
=â¡â¢â¢â£
1 0 02 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 0
â1 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 00 â1 1
â€â¥â¥âŠ
=â¡â¢â¢â£
1 0 02 1 0
â1 â1 1
â€â¥â¥âŠ
4
ð¿ð decompostion
ðž3ðž2ðž1ðŽ = ð â¹ ðŽ = ðžâ11 ðžâ1
2 ðžâ13 ð
The factor ð¿ is given by
ð¿ = ðžâ11 ðžâ1
2 ðžâ13
=â¡â¢â¢â£
1 0 02 1 00 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 0
â1 0 1
â€â¥â¥âŠ
â¡â¢â¢â£
1 0 00 1 00 â1 1
â€â¥â¥âŠ
=â¡â¢â¢â£
1 0 02 1 0
â1 â1 1
â€â¥â¥âŠ
4
ð¿ð decompostion
We found the following ð¿ð decomposition of ðŽ:
ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
= ð¿ð =â¡â¢â¢â£
1 0 02 1 0
â1 â1 1
â€â¥â¥âŠ
â¡â¢â¢â£
2 1 10 â8 â20 0 1
â€â¥â¥âŠ
.
5
Why ð¿ð decomposition?Once we have ðŽ = ð¿ð, it is simple to solve ðŽð¥ð¥ð¥ = ððð.
ðŽð¥ð¥ð¥ = ðððð¿(ðð¥ð¥ð¥) = ððð
ð¿ððð = ððð and ðð¥ð¥ð¥ = ððð.Both of the final systems are triangular and hence easily solved:
⢠ð¿ððð = ððð by forward substitution to find ððð, and then⢠ðð¥ð¥ð¥ = ððð by backward substitution to find ð¥ð¥ð¥.
ⶠExample. Solve
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
ð¥ð¥ð¥ =â¡â¢â¢â£
410â3
â€â¥â¥âŠ
.
6
Why ð¿ð decomposition?Once we have ðŽ = ð¿ð, it is simple to solve ðŽð¥ð¥ð¥ = ððð.
ðŽð¥ð¥ð¥ = ðððð¿(ðð¥ð¥ð¥) = ððð
ð¿ððð = ððð and ðð¥ð¥ð¥ = ððð.Both of the final systems are triangular and hence easily solved:
⢠ð¿ððð = ððð by forward substitution to find ððð, and then⢠ðð¥ð¥ð¥ = ððð by backward substitution to find ð¥ð¥ð¥.
ⶠExample. Solve
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
ð¥ð¥ð¥ =â¡â¢â¢â£
410â3
â€â¥â¥âŠ
.
6
Why ð¿ð decomposition?Once we have ðŽ = ð¿ð, it is simple to solve ðŽð¥ð¥ð¥ = ððð.
ðŽð¥ð¥ð¥ = ðððð¿(ðð¥ð¥ð¥) = ððð
ð¿ððð = ððð and ðð¥ð¥ð¥ = ððð.Both of the final systems are triangular and hence easily solved:
⢠ð¿ððð = ððð by forward substitution to find ððð, and then⢠ðð¥ð¥ð¥ = ððð by backward substitution to find ð¥ð¥ð¥.
ⶠExample. Solve
â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ
ð¥ð¥ð¥ =â¡â¢â¢â£
410â3
â€â¥â¥âŠ
.
6
The inverse of a matrix
ⶠDefinition. An ð à ð matrix ðŽ is invertible if there is a matrix ðµsuch that
ðŽðµ = ðµðŽ = ðŒðÃð.
In that case, ðµ is the inverse of ðŽ and is denoted by ðŽâ1.
ⶠRemark.
⢠The inverse of a matrix is unique. (Why?)⢠Do not write ðŽ
ðµ .
7
The inverse of a matrix
ⶠDefinition. An ð à ð matrix ðŽ is invertible if there is a matrix ðµsuch that
ðŽðµ = ðµðŽ = ðŒðÃð.
In that case, ðµ is the inverse of ðŽ and is denoted by ðŽâ1.ⶠRemark.
⢠The inverse of a matrix is unique. (Why?)
⢠Do not write ðŽðµ .
7
The inverse of a matrix
ⶠDefinition. An ð à ð matrix ðŽ is invertible if there is a matrix ðµsuch that
ðŽðµ = ðµðŽ = ðŒðÃð.
In that case, ðµ is the inverse of ðŽ and is denoted by ðŽâ1.ⶠRemark.
⢠The inverse of a matrix is unique. (Why?)⢠Do not write ðŽ
ðµ .
7
The inverse of a matrix
ⶠExample. Let ðŽ = â¡â¢â£ð ðð ð
â€â¥âŠ
. If ðð â ðð â 0, then
ðŽâ1 = 1ðð â ðð
â¡â¢â£
ð âðâð ð
â€â¥âŠ
.
ⶠExample. The matrix ðŽ = â¡â¢â£0 10 0
â€â¥âŠ
is not invertible.
ⶠExample. A 2 à 2 matrix â¡â¢â£ð ðð ð
â€â¥âŠ
is invertible if and only if
ðð â ðð â 0.
8
The inverse of a matrix
ⶠExample. Let ðŽ = â¡â¢â£ð ðð ð
â€â¥âŠ
. If ðð â ðð â 0, then
ðŽâ1 = 1ðð â ðð
â¡â¢â£
ð âðâð ð
â€â¥âŠ
.
ⶠExample. The matrix ðŽ = â¡â¢â£0 10 0
â€â¥âŠ
is not invertible.
ⶠExample. A 2 à 2 matrix â¡â¢â£ð ðð ð
â€â¥âŠ
is invertible if and only if
ðð â ðð â 0.
8
The inverse of a matrix
ⶠExample. Let ðŽ = â¡â¢â£ð ðð ð
â€â¥âŠ
. If ðð â ðð â 0, then
ðŽâ1 = 1ðð â ðð
â¡â¢â£
ð âðâð ð
â€â¥âŠ
.
ⶠExample. The matrix ðŽ = â¡â¢â£0 10 0
â€â¥âŠ
is not invertible.
ⶠExample. A 2 à 2 matrix â¡â¢â£ð ðð ð
â€â¥âŠ
is invertible if and only if
ðð â ðð â 0.
8
The inverse of a matrix
Suppose ðŽ and ðµ are invertible. Then
⢠ðŽâ1 is invertible and (ðŽâ1)â1 = ðŽ.
⢠ðŽð is invertible and (ðŽð )â1 = (ðŽâ1)ð .⢠ðŽðµ is invertible and (ðŽðµ)â1 = ðµâ1ðŽâ1. (Why?)
9
The inverse of a matrix
Suppose ðŽ and ðµ are invertible. Then
⢠ðŽâ1 is invertible and (ðŽâ1)â1 = ðŽ.⢠ðŽð is invertible and (ðŽð )â1 = (ðŽâ1)ð .
⢠ðŽðµ is invertible and (ðŽðµ)â1 = ðµâ1ðŽâ1. (Why?)
9
The inverse of a matrix
Suppose ðŽ and ðµ are invertible. Then
⢠ðŽâ1 is invertible and (ðŽâ1)â1 = ðŽ.⢠ðŽð is invertible and (ðŽð )â1 = (ðŽâ1)ð .⢠ðŽðµ is invertible and (ðŽðµ)â1 = ðµâ1ðŽâ1. (Why?)
9
Solving systems using matrix inverse
Theorem. Let ðŽ be invertible. Then the system ðŽð¥ð¥ð¥ = ððð has theunique solution ð¥ð¥ð¥ = ðŽâ1ððð.
10
Computing the inverse
ⶠTo solve ðŽð¥ð¥ð¥ = ððð, we do row reduction on [ ðŽ ⣠ð ].
ⶠTo solve ðŽð = ðŒ, we do row reduction on [ ðŽ ⣠ðŒ ].ⶠTo compute ðŽâ1 (The Gauss-Jordan Method):
⢠Form the augmented matrix [ ðŽ ⣠ðŒ ].⢠Compute the reduced echelon form.⢠If ðŽ is invertible, the result is of the form [ ðŒ ⣠ðŽâ1 ].
11
Computing the inverse
ⶠTo solve ðŽð¥ð¥ð¥ = ððð, we do row reduction on [ ðŽ ⣠ð ].ⶠTo solve ðŽð = ðŒ, we do row reduction on [ ðŽ ⣠ðŒ ].
ⶠTo compute ðŽâ1 (The Gauss-Jordan Method):
⢠Form the augmented matrix [ ðŽ ⣠ðŒ ].⢠Compute the reduced echelon form.⢠If ðŽ is invertible, the result is of the form [ ðŒ ⣠ðŽâ1 ].
11
Computing the inverse
ⶠTo solve ðŽð¥ð¥ð¥ = ððð, we do row reduction on [ ðŽ ⣠ð ].ⶠTo solve ðŽð = ðŒ, we do row reduction on [ ðŽ ⣠ðŒ ].ⶠTo compute ðŽâ1 (The Gauss-Jordan Method):
⢠Form the augmented matrix [ ðŽ ⣠ðŒ ].⢠Compute the reduced echelon form.⢠If ðŽ is invertible, the result is of the form [ ðŒ ⣠ðŽâ1 ].
11
Computing the inverse
ⶠExample. Find the inverse of ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ, if it exists.
ⶠSolution.
â¡â¢â¢â£
2 1 1 1 0 04 â6 0 0 1 0
â2 7 2 0 0 1
â€â¥â¥âŠ
ð 2â¶ð 2â2ð 1âââââââââââð 3â¶ð 3+ð 1
â¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 8 3 1 0 1
â€â¥â¥âŠ
ð 3â¶ð 3+ð 2âââââââââââ¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 0 1 â1 1 1
â€â¥â¥âŠ
âââââââ âŠ
12
Computing the inverse
ⶠExample. Find the inverse of ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ, if it exists.
ⶠSolution.
â¡â¢â¢â£
2 1 1 1 0 04 â6 0 0 1 0
â2 7 2 0 0 1
â€â¥â¥âŠ
ð 2â¶ð 2â2ð 1âââââââââââð 3â¶ð 3+ð 1
â¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 8 3 1 0 1
â€â¥â¥âŠ
ð 3â¶ð 3+ð 2âââââââââââ¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 0 1 â1 1 1
â€â¥â¥âŠ
âââââââ âŠ
12
Computing the inverse
ⶠExample. Find the inverse of ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ, if it exists.
ⶠSolution.
â¡â¢â¢â£
2 1 1 1 0 04 â6 0 0 1 0
â2 7 2 0 0 1
â€â¥â¥âŠ
ð 2â¶ð 2â2ð 1âââââââââââð 3â¶ð 3+ð 1
â¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 8 3 1 0 1
â€â¥â¥âŠ
ð 3â¶ð 3+ð 2âââââââââââ¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 0 1 â1 1 1
â€â¥â¥âŠ
âââââââ âŠ
12
Computing the inverse
ⶠExample. Find the inverse of ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ, if it exists.
ⶠSolution.
â¡â¢â¢â£
2 1 1 1 0 04 â6 0 0 1 0
â2 7 2 0 0 1
â€â¥â¥âŠ
ð 2â¶ð 2â2ð 1âââââââââââð 3â¶ð 3+ð 1
â¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 8 3 1 0 1
â€â¥â¥âŠ
ð 3â¶ð 3+ð 2âââââââââââ¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 0 1 â1 1 1
â€â¥â¥âŠ
âââââââ âŠ
12
Computing the inverse
ⶠExample. Find the inverse of ðŽ =â¡â¢â¢â£
2 1 14 â6 0
â2 7 2
â€â¥â¥âŠ, if it exists.
ⶠSolution.
â¡â¢â¢â£
2 1 1 1 0 04 â6 0 0 1 0
â2 7 2 0 0 1
â€â¥â¥âŠ
ð 2â¶ð 2â2ð 1âââââââââââð 3â¶ð 3+ð 1
â¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 8 3 1 0 1
â€â¥â¥âŠ
ð 3â¶ð 3+ð 2âââââââââââ¡â¢â¢â£
2 1 1 1 0 00 â8 â2 â2 1 00 0 1 â1 1 1
â€â¥â¥âŠ
âââââââ âŠ
12
Computing the inverse
ââââââââ¡â¢â¢â£
1 0 0 1216
â516
â616
0 1 0 48
â38
â28
0 0 1 â1 1 1
â€â¥â¥âŠ
ðŽâ1 =â¡â¢â¢â£
1216
â516
â616
48
â38
â28
â1 1 1
â€â¥â¥âŠ
13
Why does it work?
⢠Each row reduction corresponds to multiplying with anelementary matrix ðž:
[ ðŽâ£ ðŒ ] â [ ðž1ðŽâ£ ðž1ðŒ ] â [ ðž2ðž1ðŽâ£ ðž2ðž1 ] â âŠ
⊠â [ ð¹ðŽ ⣠ð¹] where ð¹ = ðžð ⊠ðž2ðž1.
⢠If we manage to reduce [ ðŽâ£ ðŒ ] to [ ðŒâ£ ð¹ ], this means
ð¹ðŽ = ðŒ â¹ ðŽâ1 = ð¹.
14
Why does it work?
⢠Each row reduction corresponds to multiplying with anelementary matrix ðž:
[ ðŽâ£ ðŒ ] â [ ðž1ðŽâ£ ðž1ðŒ ] â [ ðž2ðž1ðŽâ£ ðž2ðž1 ] â âŠ
⊠â [ ð¹ðŽ ⣠ð¹] where ð¹ = ðžð ⊠ðž2ðž1.
⢠If we manage to reduce [ ðŽâ£ ðŒ ] to [ ðŒâ£ ð¹ ], this means
ð¹ðŽ = ðŒ â¹ ðŽâ1 = ð¹.
14
Why does it work?
⢠Each row reduction corresponds to multiplying with anelementary matrix ðž:
[ ðŽâ£ ðŒ ] â [ ðž1ðŽâ£ ðž1ðŒ ] â [ ðž2ðž1ðŽâ£ ðž2ðž1 ] â âŠ
⊠â [ ð¹ðŽ ⣠ð¹] where ð¹ = ðžð ⊠ðž2ðž1.
⢠If we manage to reduce [ ðŽâ£ ðŒ ] to [ ðŒâ£ ð¹ ], this means
ð¹ðŽ = ðŒ â¹ ðŽâ1 = ð¹.
14
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