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How to find a four square-equations matrix with only one known equation?
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Sorry if the title isn't very descriptive. However, I've seen the following problem while I'm solving linear problems
Write down $E=||Ax-b||^2$ as a sum of four squares - the last one is $(C+4D-20)^2$. Find the derivative equations $partial E/partial C=0$ and $partial E/partial D=0$. Divide by $2$ to obtain the normal equations $A^TAhatx=A^Tb$.
with the given solution
Observe $$A=beginpmatrix1 &0\1 &1\1 &3\1 &4\endpmatrixtext, b=beginpmatrix0\8\8\20endpmatrixtext , and define x=beginpmatrixC\Dendpmatrix$$
$$Ax-b=beginpmatrixC\C+D-8\C + 3D − 8\C + 4D − 20\endpmatrixtext, $$
$||Ax − b||^2 = C^2 + (C + D − 8)^2 + (C + 3D − 8)^2 + (C + 4D − 20)^2$
.
Where do the values of the matrix A come from? I don't find any relative thing in the question. Thanks.
linear-algebra numerical-linear-algebra
asked 2 days ago
Dia Abujaber
13
add a comment |Â
up vote
0
down vote
favorite
Sorry if the title isn't very descriptive. However, I've seen the following problem while I'm solving linear problems
Write down $E=||Ax-b||^2$ as a sum of four squares - the last one is $(C+4D-20)^2$. Find the derivative equations $partial E/partial C=0$ and $partial E/partial D=0$. Divide by $2$ to obtain the normal equations $A^TAhatx=A^Tb$.
with the given solution
Observe $$A=beginpmatrix1 &0\1 &1\1 &3\1 &4\endpmatrixtext, b=beginpmatrix0\8\8\20endpmatrixtext , and define x=beginpmatrixC\Dendpmatrix$$
$$Ax-b=beginpmatrixC\C+D-8\C + 3D − 8\C + 4D − 20\endpmatrixtext, $$
$||Ax − b||^2 = C^2 + (C + D − 8)^2 + (C + 3D − 8)^2 + (C + 4D − 20)^2$
.
Where do the values of the matrix A come from? I don't find any relative thing in the question. Thanks.
linear-algebra numerical-linear-algebra
asked 2 days ago
Dia Abujaber
13
What have you worked out so far ?
– Ahmad Bazzi
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Sorry if the title isn't very descriptive. However, I've seen the following problem while I'm solving linear problems
Write down $E=||Ax-b||^2$ as a sum of four squares - the last one is $(C+4D-20)^2$. Find the derivative equations $partial E/partial C=0$ and $partial E/partial D=0$. Divide by $2$ to obtain the normal equations $A^TAhatx=A^Tb$.
with the given solution
Observe $$A=beginpmatrix1 &0\1 &1\1 &3\1 &4\endpmatrixtext, b=beginpmatrix0\8\8\20endpmatrixtext , and define x=beginpmatrixC\Dendpmatrix$$
$$Ax-b=beginpmatrixC\C+D-8\C + 3D − 8\C + 4D − 20\endpmatrixtext, $$
$||Ax − b||^2 = C^2 + (C + D − 8)^2 + (C + 3D − 8)^2 + (C + 4D − 20)^2$
.
Where do the values of the matrix A come from? I don't find any relative thing in the question. Thanks.
linear-algebra numerical-linear-algebra
asked 2 days ago
Dia Abujaber
13
Sorry if the title isn't very descriptive. However, I've seen the following problem while I'm solving linear problems
Write down $E=||Ax-b||^2$ as a sum of four squares - the last one is $(C+4D-20)^2$. Find the derivative equations $partial E/partial C=0$ and $partial E/partial D=0$. Divide by $2$ to obtain the normal equations $A^TAhatx=A^Tb$.
with the given solution
Observe $$A=beginpmatrix1 &0\1 &1\1 &3\1 &4\endpmatrixtext, b=beginpmatrix0\8\8\20endpmatrixtext , and define x=beginpmatrixC\Dendpmatrix$$
$$Ax-b=beginpmatrixC\C+D-8\C + 3D − 8\C + 4D − 20\endpmatrixtext, $$
$||Ax − b||^2 = C^2 + (C + D − 8)^2 + (C + 3D − 8)^2 + (C + 4D − 20)^2$
.
Where do the values of the matrix A come from? I don't find any relative thing in the question. Thanks.
linear-algebra numerical-linear-algebra
asked 2 days ago
Dia Abujaber
13
edited 2 days ago
asked 2 days ago
Dia Abujaber
13
asked 2 days ago
Dia Abujaber
13
asked 2 days ago
Dia Abujaber
13
13
What have you worked out so far ?
– Ahmad Bazzi
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago
add a comment |Â
What have you worked out so far ?
– Ahmad Bazzi
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago
What have you worked out so far ?
– Ahmad Bazzi
2 days ago
What have you worked out so far ?
– Ahmad Bazzi
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago
add a comment |Â
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What have you worked out so far ?
– Ahmad Bazzi
2 days ago
i believe in your notations $A$ is the feature-matrix and $b$ is your observations. They are given in order to obtain an estimate for your vector $x$, i.e. to compute $hatx$.
– pointguard0
2 days ago
@pointguard0 Yea, but how did we get both A and b values, they were given in the solution.
– Dia Abujaber
2 days ago
you cannot minimze $| Ax - b|_2^2$ without having observed values of $A$ and $b$, please edit your question adding the complete formulation of your task.
– pointguard0
2 days ago
I guess there's a mistake in the problem itself formulation from the original source
– Dia Abujaber
2 days ago