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The decomposition for a symmetric positiv definite matrix is unique
$begingroup$
We have the matrix begin{equation*}A=begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \ 1/5 & 1/2 & 1/5 & 1/10 \ 1/10 & 1/5 & 1/2 & 1/5 \ 1/17 & 1/10 & 1/5 & 1/10end{pmatrix}end{equation*}
I have applied the Cholesky decomposition and found that $A=tilde{L}cdot tilde{L}^T$ where
begin{equation*}tilde{L}=begin{pmatrix}frac{1}{sqrt{2}} & 0 & 0 & 0 \ frac{sqrt{2}}{5} & sqrt{frac{21}{50}} & 0 & 0 \ frac{sqrt{2}}{10} & frac{4sqrt{42}}{105} & frac{2sqrt{1155}}{105} & 0 \ frac{sqrt{2}}{17} & frac{13}{17sqrt{42}} & frac{142}{17sqrt{1155}} & sqrt{frac{298}{15895}}end{pmatrix} text{ und } tilde{L}^T=begin{pmatrix}frac{1}{sqrt{2}} & frac{sqrt{2}}{5} & frac{sqrt{2}}{10} & frac{sqrt{2}}{17} \ 0 & sqrt{frac{21}{50}} & frac{4sqrt{42}}{105} & frac{13}{17sqrt{42}} \ 0 & 0 & frac{2sqrt{1155}}{105} & frac{142}{17sqrt{1155}} \ 0 & 0 & 0 & sqrt{frac{298}{15895}} end{pmatrix}end{equation*}
Is that correct?
I want to show that the decomposition for a symmetric positiv definite matrix is unique and it is given as a hint that we have to use the LU decomposition.
Could you explain to me how we can do that?
matrices matrix-decomposition lu-decomposition
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
$endgroup$
add a comment |
$begingroup$
We have the matrix begin{equation*}A=begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \ 1/5 & 1/2 & 1/5 & 1/10 \ 1/10 & 1/5 & 1/2 & 1/5 \ 1/17 & 1/10 & 1/5 & 1/10end{pmatrix}end{equation*}
I have applied the Cholesky decomposition and found that $A=tilde{L}cdot tilde{L}^T$ where
begin{equation*}tilde{L}=begin{pmatrix}frac{1}{sqrt{2}} & 0 & 0 & 0 \ frac{sqrt{2}}{5} & sqrt{frac{21}{50}} & 0 & 0 \ frac{sqrt{2}}{10} & frac{4sqrt{42}}{105} & frac{2sqrt{1155}}{105} & 0 \ frac{sqrt{2}}{17} & frac{13}{17sqrt{42}} & frac{142}{17sqrt{1155}} & sqrt{frac{298}{15895}}end{pmatrix} text{ und } tilde{L}^T=begin{pmatrix}frac{1}{sqrt{2}} & frac{sqrt{2}}{5} & frac{sqrt{2}}{10} & frac{sqrt{2}}{17} \ 0 & sqrt{frac{21}{50}} & frac{4sqrt{42}}{105} & frac{13}{17sqrt{42}} \ 0 & 0 & frac{2sqrt{1155}}{105} & frac{142}{17sqrt{1155}} \ 0 & 0 & 0 & sqrt{frac{298}{15895}} end{pmatrix}end{equation*}
Is that correct?
I want to show that the decomposition for a symmetric positiv definite matrix is unique and it is given as a hint that we have to use the LU decomposition.
Could you explain to me how we can do that?
matrices matrix-decomposition lu-decomposition
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
$endgroup$
add a comment |
$begingroup$
We have the matrix begin{equation*}A=begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \ 1/5 & 1/2 & 1/5 & 1/10 \ 1/10 & 1/5 & 1/2 & 1/5 \ 1/17 & 1/10 & 1/5 & 1/10end{pmatrix}end{equation*}
I have applied the Cholesky decomposition and found that $A=tilde{L}cdot tilde{L}^T$ where
begin{equation*}tilde{L}=begin{pmatrix}frac{1}{sqrt{2}} & 0 & 0 & 0 \ frac{sqrt{2}}{5} & sqrt{frac{21}{50}} & 0 & 0 \ frac{sqrt{2}}{10} & frac{4sqrt{42}}{105} & frac{2sqrt{1155}}{105} & 0 \ frac{sqrt{2}}{17} & frac{13}{17sqrt{42}} & frac{142}{17sqrt{1155}} & sqrt{frac{298}{15895}}end{pmatrix} text{ und } tilde{L}^T=begin{pmatrix}frac{1}{sqrt{2}} & frac{sqrt{2}}{5} & frac{sqrt{2}}{10} & frac{sqrt{2}}{17} \ 0 & sqrt{frac{21}{50}} & frac{4sqrt{42}}{105} & frac{13}{17sqrt{42}} \ 0 & 0 & frac{2sqrt{1155}}{105} & frac{142}{17sqrt{1155}} \ 0 & 0 & 0 & sqrt{frac{298}{15895}} end{pmatrix}end{equation*}
Is that correct?
I want to show that the decomposition for a symmetric positiv definite matrix is unique and it is given as a hint that we have to use the LU decomposition.
Could you explain to me how we can do that?
matrices matrix-decomposition lu-decomposition
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
$endgroup$
We have the matrix begin{equation*}A=begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \ 1/5 & 1/2 & 1/5 & 1/10 \ 1/10 & 1/5 & 1/2 & 1/5 \ 1/17 & 1/10 & 1/5 & 1/10end{pmatrix}end{equation*}
I have applied the Cholesky decomposition and found that $A=tilde{L}cdot tilde{L}^T$ where
begin{equation*}tilde{L}=begin{pmatrix}frac{1}{sqrt{2}} & 0 & 0 & 0 \ frac{sqrt{2}}{5} & sqrt{frac{21}{50}} & 0 & 0 \ frac{sqrt{2}}{10} & frac{4sqrt{42}}{105} & frac{2sqrt{1155}}{105} & 0 \ frac{sqrt{2}}{17} & frac{13}{17sqrt{42}} & frac{142}{17sqrt{1155}} & sqrt{frac{298}{15895}}end{pmatrix} text{ und } tilde{L}^T=begin{pmatrix}frac{1}{sqrt{2}} & frac{sqrt{2}}{5} & frac{sqrt{2}}{10} & frac{sqrt{2}}{17} \ 0 & sqrt{frac{21}{50}} & frac{4sqrt{42}}{105} & frac{13}{17sqrt{42}} \ 0 & 0 & frac{2sqrt{1155}}{105} & frac{142}{17sqrt{1155}} \ 0 & 0 & 0 & sqrt{frac{298}{15895}} end{pmatrix}end{equation*}
Is that correct?
I want to show that the decomposition for a symmetric positiv definite matrix is unique and it is given as a hint that we have to use the LU decomposition.
Could you explain to me how we can do that?
matrices matrix-decomposition lu-decomposition
matrices matrix-decomposition lu-decomposition
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
asked Jan 31 at 19:14
Mary StarMary Star
3,10982476
3,10982476
add a comment |
add a comment |
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