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Applied Linear Algebra. Review. LU Factorization. If a square matrix can strict upper triangular form, U, without interchanging any rows, then A can be factored as A=LU, where L is a low triangular matrix. Ax= b L(Ux )= b Solve Ly= b then y = Ux. Elementary Matrices. - PowerPoint PPT Presentation
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Applied Linear Algebra
Review
LU Factorization
• If a square matrix can strict upper triangular form, U, without interchanging any rows, then A can be factored as A=LU, where L is a low triangular matrix.
Ax=bL(Ux)=b
Solve Ly=b then y=Ux
Elementary Matrices
I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I
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E1A =
0 1 0
1 0 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
AE1 =
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
0 1 0
1 0 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
a21 a22 a23
a11 a12 a13
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
a12 a11 a13
a22 a21 a23
a32 a31 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Row 1 and Row 2 are switched!
Column 1 and Column 2 are switched!
Elementary Matrices
Elementary Matrices
I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I
II. An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant
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E2A =
1 0 0
0 3 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
AE2 =
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
1 0 0
0 3 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
a11 a12 a13
3a21 3a22 3a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Row 2 is multiplied by 3!
Column 2 is multiplied by 3!
€
a11 3a12 a13
a21 3a22 a23
a31 3a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Elementary Matrices
Elementary Matrices
I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I
II. An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant.
III. An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.
€
E3A =
1 0 4
0 1 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
AE3 =
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥⋅
1 0 4
0 1 0
0 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥=
€
a11 + 4a31 a12 + 4a32 a13 + 4a33
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
4 times Row 3 is added to Row 1
4 times Column 1 is added to column 3
€
a11 a12 4a11+a13
a21 a22 4a21+a23
a31 a32 4a31+a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Elementary Matrices
Change of Basis
Change of Basis
Gram-Schmidt Orthonomalization Process
