Uniqueness of a solution of a matrix
Given $A in C^{mtimes n}$ of rank $n$ and $b in C^{m}$, now consider the below system of equation
$$begin{equation}
r + Ax = b \
A^*r = 0
end{equation}$$
where $I$ is $m times m$ identity matrix. How can I show that $(r,x)^T$ is the unique solution where $r$ and $x$ are the residual and solution of below problem? $$minimize : ||Ax-b||_2 $$ This is a question from Numerical Algebra book by Lloyd.
linear-algebra matrices matrix-equations numerical-linear-algebra
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
asked Oct 24 at 2:57
user3375977
11
add a comment |
Given $A in C^{mtimes n}$ of rank $n$ and $b in C^{m}$, now consider the below system of equation
$$begin{equation}
r + Ax = b \
A^*r = 0
end{equation}$$
where $I$ is $m times m$ identity matrix. How can I show that $(r,x)^T$ is the unique solution where $r$ and $x$ are the residual and solution of below problem? $$minimize : ||Ax-b||_2 $$ This is a question from Numerical Algebra book by Lloyd.
linear-algebra matrices matrix-equations numerical-linear-algebra
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
asked Oct 24 at 2:57
user3375977
11
add a comment |
Given $A in C^{mtimes n}$ of rank $n$ and $b in C^{m}$, now consider the below system of equation
$$begin{equation}
r + Ax = b \
A^*r = 0
end{equation}$$
where $I$ is $m times m$ identity matrix. How can I show that $(r,x)^T$ is the unique solution where $r$ and $x$ are the residual and solution of below problem? $$minimize : ||Ax-b||_2 $$ This is a question from Numerical Algebra book by Lloyd.
linear-algebra matrices matrix-equations numerical-linear-algebra
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
asked Oct 24 at 2:57
user3375977
11
Given $A in C^{mtimes n}$ of rank $n$ and $b in C^{m}$, now consider the below system of equation
$$begin{equation}
r + Ax = b \
A^*r = 0
end{equation}$$
where $I$ is $m times m$ identity matrix. How can I show that $(r,x)^T$ is the unique solution where $r$ and $x$ are the residual and solution of below problem? $$minimize : ||Ax-b||_2 $$ This is a question from Numerical Algebra book by Lloyd.
linear-algebra matrices matrix-equations numerical-linear-algebra
linear-algebra matrices matrix-equations numerical-linear-algebra
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
asked Oct 24 at 2:57
user3375977
11
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
asked Oct 24 at 2:57
user3375977
11
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
edited Nov 24 at 19:51
Mostafa Ayaz
13.7k3836
13.7k3836
asked Oct 24 at 2:57
user3375977
11
asked Oct 24 at 2:57
user3375977
11
asked Oct 24 at 2:57
user3375977
11
11
add a comment |
add a comment |
1 Answer
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oldest
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Denote by $mathcal C$ the column space of $A$. Then $mathcal C^perp=left{w: A^ast w=0right}$. Let $b=Au+v$ where $vinmathcal C^perp$. Then $|Ax-b|^2 = |A(x-u)|^2 + |v|^2$. Hence $|Ax-b|$ is minimised if and only if $|A(x-u)|$ is minimised, and your problem boils down to showing that the equation
$$
A(x-u) + (r-v) = 0,quad rinmathcal C^perp
$$
has a unique solution $(x,r)$ and this $x$ is also the unique global minimiser of $|A(x-u)|$. The rest should be straightforward.
answered Oct 24 at 9:46
user1551
71.2k566125
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
Denote by $mathcal C$ the column space of $A$. Then $mathcal C^perp=left{w: A^ast w=0right}$. Let $b=Au+v$ where $vinmathcal C^perp$. Then $|Ax-b|^2 = |A(x-u)|^2 + |v|^2$. Hence $|Ax-b|$ is minimised if and only if $|A(x-u)|$ is minimised, and your problem boils down to showing that the equation
$$
A(x-u) + (r-v) = 0,quad rinmathcal C^perp
$$
has a unique solution $(x,r)$ and this $x$ is also the unique global minimiser of $|A(x-u)|$. The rest should be straightforward.
answered Oct 24 at 9:46
user1551
71.2k566125
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
add a comment |
Denote by $mathcal C$ the column space of $A$. Then $mathcal C^perp=left{w: A^ast w=0right}$. Let $b=Au+v$ where $vinmathcal C^perp$. Then $|Ax-b|^2 = |A(x-u)|^2 + |v|^2$. Hence $|Ax-b|$ is minimised if and only if $|A(x-u)|$ is minimised, and your problem boils down to showing that the equation
$$
A(x-u) + (r-v) = 0,quad rinmathcal C^perp
$$
has a unique solution $(x,r)$ and this $x$ is also the unique global minimiser of $|A(x-u)|$. The rest should be straightforward.
answered Oct 24 at 9:46
user1551
71.2k566125
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
add a comment |
Denote by $mathcal C$ the column space of $A$. Then $mathcal C^perp=left{w: A^ast w=0right}$. Let $b=Au+v$ where $vinmathcal C^perp$. Then $|Ax-b|^2 = |A(x-u)|^2 + |v|^2$. Hence $|Ax-b|$ is minimised if and only if $|A(x-u)|$ is minimised, and your problem boils down to showing that the equation
$$
A(x-u) + (r-v) = 0,quad rinmathcal C^perp
$$
has a unique solution $(x,r)$ and this $x$ is also the unique global minimiser of $|A(x-u)|$. The rest should be straightforward.
answered Oct 24 at 9:46
user1551
71.2k566125
Denote by $mathcal C$ the column space of $A$. Then $mathcal C^perp=left{w: A^ast w=0right}$. Let $b=Au+v$ where $vinmathcal C^perp$. Then $|Ax-b|^2 = |A(x-u)|^2 + |v|^2$. Hence $|Ax-b|$ is minimised if and only if $|A(x-u)|$ is minimised, and your problem boils down to showing that the equation
$$
A(x-u) + (r-v) = 0,quad rinmathcal C^perp
$$
has a unique solution $(x,r)$ and this $x$ is also the unique global minimiser of $|A(x-u)|$. The rest should be straightforward.
answered Oct 24 at 9:46
user1551
71.2k566125
answered Oct 24 at 9:46
user1551
71.2k566125
answered Oct 24 at 9:46
user1551
71.2k566125
answered Oct 24 at 9:46
user1551
71.2k566125
71.2k566125
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
add a comment |
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
Can you please elaborate?
– Dushyant Sahoo
Oct 30 at 7:34
add a comment |
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