Compare classical and modified algorithm of Cholesky
$begingroup$
Let $textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $textbf{A} = textbf{R}^Ttextbf{D}textbf{R}$ and the classical factorization $textbf{A} = textbf{R}_c^Ttextbf{R}_c$
Given $textbf{R}$ and $textbf{D}$, determine $textbf{R}_c$
$textbf{Proof}$:
Let
$$
textbf{D}^{1/2} = begin{bmatrix}
sqrt{d_1} &&& \
&sqrt{d_2}&&\
&&ddots&\
&&&sqrt{d_n}
end{bmatrix}
$$
Then multiply each row in the modified Cholesky matrix textbf{R} by $sqrt{d_{i,i}}$ to obtain the classical Cholesky matrix
$$
textbf{R}_c = textbf{D}^{1/2}textbf{R}
$$
This is confirmed by
begin{align}
textbf{R}_c^Ttextbf{R}_c &= (textbf{D}^{1/2}textbf{R})^T(textbf{D}^{1/2}textbf{R})tag{1}label{eq1} \
&= textbf{R}^Ttextbf{D}^{1/2}textbf{D}^{1/2}textbf{R}tag{2}label{eq2} \&= textbf{R}^Ttextbf{D}textbf{R}tag{3}label{eq3}
end{align}
What is the name of the property of matrix that allow to go from (1) to (2)
linear-algebra diagonalization symmetry transpose
asked Dec 27 '18 at 18:41
ecjbecjb
28718
$endgroup$
add a comment |
$begingroup$
Let $textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $textbf{A} = textbf{R}^Ttextbf{D}textbf{R}$ and the classical factorization $textbf{A} = textbf{R}_c^Ttextbf{R}_c$
Given $textbf{R}$ and $textbf{D}$, determine $textbf{R}_c$
$textbf{Proof}$:
Let
$$
textbf{D}^{1/2} = begin{bmatrix}
sqrt{d_1} &&& \
&sqrt{d_2}&&\
&&ddots&\
&&&sqrt{d_n}
end{bmatrix}
$$
Then multiply each row in the modified Cholesky matrix textbf{R} by $sqrt{d_{i,i}}$ to obtain the classical Cholesky matrix
$$
textbf{R}_c = textbf{D}^{1/2}textbf{R}
$$
This is confirmed by
begin{align}
textbf{R}_c^Ttextbf{R}_c &= (textbf{D}^{1/2}textbf{R})^T(textbf{D}^{1/2}textbf{R})tag{1}label{eq1} \
&= textbf{R}^Ttextbf{D}^{1/2}textbf{D}^{1/2}textbf{R}tag{2}label{eq2} \&= textbf{R}^Ttextbf{D}textbf{R}tag{3}label{eq3}
end{align}
What is the name of the property of matrix that allow to go from (1) to (2)
linear-algebra diagonalization symmetry transpose
asked Dec 27 '18 at 18:41
ecjbecjb
28718
$endgroup$
1
$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43
add a comment |
$begingroup$
Let $textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $textbf{A} = textbf{R}^Ttextbf{D}textbf{R}$ and the classical factorization $textbf{A} = textbf{R}_c^Ttextbf{R}_c$
Given $textbf{R}$ and $textbf{D}$, determine $textbf{R}_c$
$textbf{Proof}$:
Let
$$
textbf{D}^{1/2} = begin{bmatrix}
sqrt{d_1} &&& \
&sqrt{d_2}&&\
&&ddots&\
&&&sqrt{d_n}
end{bmatrix}
$$
Then multiply each row in the modified Cholesky matrix textbf{R} by $sqrt{d_{i,i}}$ to obtain the classical Cholesky matrix
$$
textbf{R}_c = textbf{D}^{1/2}textbf{R}
$$
This is confirmed by
begin{align}
textbf{R}_c^Ttextbf{R}_c &= (textbf{D}^{1/2}textbf{R})^T(textbf{D}^{1/2}textbf{R})tag{1}label{eq1} \
&= textbf{R}^Ttextbf{D}^{1/2}textbf{D}^{1/2}textbf{R}tag{2}label{eq2} \&= textbf{R}^Ttextbf{D}textbf{R}tag{3}label{eq3}
end{align}
What is the name of the property of matrix that allow to go from (1) to (2)
linear-algebra diagonalization symmetry transpose
asked Dec 27 '18 at 18:41
ecjbecjb
28718
$endgroup$
Let $textbf{A}$ be a symmetric matrix with tthe modified Cholesky factorization $textbf{A} = textbf{R}^Ttextbf{D}textbf{R}$ and the classical factorization $textbf{A} = textbf{R}_c^Ttextbf{R}_c$
Given $textbf{R}$ and $textbf{D}$, determine $textbf{R}_c$
$textbf{Proof}$:
Let
$$
textbf{D}^{1/2} = begin{bmatrix}
sqrt{d_1} &&& \
&sqrt{d_2}&&\
&&ddots&\
&&&sqrt{d_n}
end{bmatrix}
$$
Then multiply each row in the modified Cholesky matrix textbf{R} by $sqrt{d_{i,i}}$ to obtain the classical Cholesky matrix
$$
textbf{R}_c = textbf{D}^{1/2}textbf{R}
$$
This is confirmed by
begin{align}
textbf{R}_c^Ttextbf{R}_c &= (textbf{D}^{1/2}textbf{R})^T(textbf{D}^{1/2}textbf{R})tag{1}label{eq1} \
&= textbf{R}^Ttextbf{D}^{1/2}textbf{D}^{1/2}textbf{R}tag{2}label{eq2} \&= textbf{R}^Ttextbf{D}textbf{R}tag{3}label{eq3}
end{align}
What is the name of the property of matrix that allow to go from (1) to (2)
linear-algebra diagonalization symmetry transpose
linear-algebra diagonalization symmetry transpose
asked Dec 27 '18 at 18:41
ecjbecjb
28718
asked Dec 27 '18 at 18:41
ecjbecjb
28718
asked Dec 27 '18 at 18:41
ecjbecjb
28718
asked Dec 27 '18 at 18:41
ecjbecjb
28718
asked Dec 27 '18 at 18:41
ecjbecjb
28718
28718
1
$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43
add a comment |
1
$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43
1
1
$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43
$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43
add a comment |
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$begingroup$
Distributing a transpose causes order of matrix multiplication to reverse. It's similar to inverse. Also transposing a diagonal matrix does nothing.
$endgroup$
– Mark
Dec 27 '18 at 18:43