Alternatives in Farka's Lemma as boudaries
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I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas.
Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Then $lim_{x rightarrow x^*} y(x)=infty$, where $infty$ is a vector where at least one entry is equal to infinity.
Lemma 2: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Let $z$ have no zero entries. Then $lim_{x→x^*}y(x)=infty^{*}$, where $∞^*$ is a vector where all entries are equal to infinity.
I have look on many books to find these theorems or a perfected form of them, with no results. Can someone help?
matrices matrix-equations matrix-calculus
edited Jan 9 at 2:38
Gaby Alfonso
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
$endgroup$
add a comment |
$begingroup$
I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas.
Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Then $lim_{x rightarrow x^*} y(x)=infty$, where $infty$ is a vector where at least one entry is equal to infinity.
Lemma 2: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Let $z$ have no zero entries. Then $lim_{x→x^*}y(x)=infty^{*}$, where $∞^*$ is a vector where all entries are equal to infinity.
I have look on many books to find these theorems or a perfected form of them, with no results. Can someone help?
matrices matrix-equations matrix-calculus
edited Jan 9 at 2:38
Gaby Alfonso
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
$endgroup$
$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59
add a comment |
$begingroup$
I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas.
Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Then $lim_{x rightarrow x^*} y(x)=infty$, where $infty$ is a vector where at least one entry is equal to infinity.
Lemma 2: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Let $z$ have no zero entries. Then $lim_{x→x^*}y(x)=infty^{*}$, where $∞^*$ is a vector where all entries are equal to infinity.
I have look on many books to find these theorems or a perfected form of them, with no results. Can someone help?
matrices matrix-equations matrix-calculus
edited Jan 9 at 2:38
Gaby Alfonso
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
$endgroup$
I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas.
Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Then $lim_{x rightarrow x^*} y(x)=infty$, where $infty$ is a vector where at least one entry is equal to infinity.
Lemma 2: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Let $z$ have no zero entries. Then $lim_{x→x^*}y(x)=infty^{*}$, where $∞^*$ is a vector where all entries are equal to infinity.
I have look on many books to find these theorems or a perfected form of them, with no results. Can someone help?
matrices matrix-equations matrix-calculus
matrices matrix-equations matrix-calculus
edited Jan 9 at 2:38
Gaby Alfonso
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
edited Jan 9 at 2:38
Gaby Alfonso
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
edited Jan 9 at 2:38
Gaby Alfonso
697315
edited Jan 9 at 2:38
Gaby Alfonso
697315
edited Jan 9 at 2:38
Gaby Alfonso
697315
697315
asked Jan 9 at 1:40
a.giannela.giannel
162
asked Jan 9 at 1:40
a.giannela.giannel
162
asked Jan 9 at 1:40
a.giannela.giannel
162
162
$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59
add a comment |
$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59
$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59
add a comment |
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$begingroup$
What exactly is the question?
$endgroup$
– littleO
Jan 9 at 1:51
$begingroup$
The question is whether you have seen this before and, if not, if it looks right to you.
$endgroup$
– a.giannel
Jan 9 at 1:59