Mixing
Are matrices which yield a given characteristic polynomial and have specified structure connected?
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Let us consider a subset $S$ of $M_4(mathbb R)$ which has following form
beginalign*
beginpmatrix
0 & * & 0 & * \
1 & * & 0 & * \
0 & * & 0 & * \
0 & * & 1 & *
endpmatrix,
endalign*
where $*$ can assume any real number. It is also clear for any monic $4^th$ degree real polynomial, we can at least find one realization in $S$ since the upper left block and lower right block can be considered as in companion form. Let $f: S to mathbb R^n$ be the map sending the coefficients of characteristic polynomial to $mathbb R^n$.
My question is: suppose we have a square-free polynomial $p(t) = t^4 + a_3 t^3 + a_2 t^2 + a_1 t + a_0$, is $f^-1(a)$ a connected set in $M_n(mathbb R)$ where $a=(a_3, a_2, a_1, a_0)$? If we let $C$ denote the companion form of $p(t)$, which is an element of $S$, then $$f^-1(a) = V C V^-1: V in GL_4(mathbb R), V C V^-1 text is in above form .$$
I have a feeling we can first choose a realization in block form,
beginalign*
beginpmatrix
0 & * & 0 & 0 \
1 & * & 0 & 0 \
0 & 0 & 0 & * \
0 & 0 & 1 & *
endpmatrix,
endalign*
and then continuously change everything in $f^-1(a)$ into this form without changing the characteristic polynomial. Indeed, if we consider a matrix
beginalign*
A = beginpmatrix
0 & -b_1 & 0 & -c_1 \
1 & -b_2 & 0 & -c_2 \
0 & -b_3 & 0 & -c_3 \
0 & -b_4 & 1 & -c_4 \
endpmatrix,
endalign*
then the charateristic polynomial is
beginalign*
t^4 + (c_4+b_2)t^3 + (c_3 + b_1 + b_2c_4 - b_4 c_3)t^2 + (b_1 c_4 - b_3 c_3 + b_2 c_3 - b_4 c_1) t + (b_1 c_3 - b_3 c_1).
endalign*
I am thinking we should be able to continuously decrease $b_3, b_4, c_1, c_2$ to $0$ while keeping the polynomial unchanged by changing $b_1, b_2, c_3, c_4$ accordingly.
Edit: Many thanks to user9527 for putting bounty on my question.
linear-algebra general-topology matrices polynomials connectedness
asked Aug 2 at 18:07
MyCindy2012
335
This question has an open bounty worth +100
reputation from user9527 ending ending at 2018-08-12 06:45:27Z">in 4 days.
This question has not received enough attention.
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up vote
3
down vote
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Let us consider a subset $S$ of $M_4(mathbb R)$ which has following form
beginalign*
beginpmatrix
0 & * & 0 & * \
1 & * & 0 & * \
0 & * & 0 & * \
0 & * & 1 & *
endpmatrix,
endalign*
where $*$ can assume any real number. It is also clear for any monic $4^th$ degree real polynomial, we can at least find one realization in $S$ since the upper left block and lower right block can be considered as in companion form. Let $f: S to mathbb R^n$ be the map sending the coefficients of characteristic polynomial to $mathbb R^n$.
My question is: suppose we have a square-free polynomial $p(t) = t^4 + a_3 t^3 + a_2 t^2 + a_1 t + a_0$, is $f^-1(a)$ a connected set in $M_n(mathbb R)$ where $a=(a_3, a_2, a_1, a_0)$? If we let $C$ denote the companion form of $p(t)$, which is an element of $S$, then $$f^-1(a) = V C V^-1: V in GL_4(mathbb R), V C V^-1 text is in above form .$$
I have a feeling we can first choose a realization in block form,
beginalign*
beginpmatrix
0 & * & 0 & 0 \
1 & * & 0 & 0 \
0 & 0 & 0 & * \
0 & 0 & 1 & *
endpmatrix,
endalign*
and then continuously change everything in $f^-1(a)$ into this form without changing the characteristic polynomial. Indeed, if we consider a matrix
beginalign*
A = beginpmatrix
0 & -b_1 & 0 & -c_1 \
1 & -b_2 & 0 & -c_2 \
0 & -b_3 & 0 & -c_3 \
0 & -b_4 & 1 & -c_4 \
endpmatrix,
endalign*
then the charateristic polynomial is
beginalign*
t^4 + (c_4+b_2)t^3 + (c_3 + b_1 + b_2c_4 - b_4 c_3)t^2 + (b_1 c_4 - b_3 c_3 + b_2 c_3 - b_4 c_1) t + (b_1 c_3 - b_3 c_1).
endalign*
I am thinking we should be able to continuously decrease $b_3, b_4, c_1, c_2$ to $0$ while keeping the polynomial unchanged by changing $b_1, b_2, c_3, c_4$ accordingly.
Edit: Many thanks to user9527 for putting bounty on my question.
linear-algebra general-topology matrices polynomials connectedness
asked Aug 2 at 18:07
MyCindy2012
335
This question has an open bounty worth +100
reputation from user9527 ending ending at 2018-08-12 06:45:27Z">in 4 days.
This question has not received enough attention.
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let us consider a subset $S$ of $M_4(mathbb R)$ which has following form
beginalign*
beginpmatrix
0 & * & 0 & * \
1 & * & 0 & * \
0 & * & 0 & * \
0 & * & 1 & *
endpmatrix,
endalign*
where $*$ can assume any real number. It is also clear for any monic $4^th$ degree real polynomial, we can at least find one realization in $S$ since the upper left block and lower right block can be considered as in companion form. Let $f: S to mathbb R^n$ be the map sending the coefficients of characteristic polynomial to $mathbb R^n$.
My question is: suppose we have a square-free polynomial $p(t) = t^4 + a_3 t^3 + a_2 t^2 + a_1 t + a_0$, is $f^-1(a)$ a connected set in $M_n(mathbb R)$ where $a=(a_3, a_2, a_1, a_0)$? If we let $C$ denote the companion form of $p(t)$, which is an element of $S$, then $$f^-1(a) = V C V^-1: V in GL_4(mathbb R), V C V^-1 text is in above form .$$
I have a feeling we can first choose a realization in block form,
beginalign*
beginpmatrix
0 & * & 0 & 0 \
1 & * & 0 & 0 \
0 & 0 & 0 & * \
0 & 0 & 1 & *
endpmatrix,
endalign*
and then continuously change everything in $f^-1(a)$ into this form without changing the characteristic polynomial. Indeed, if we consider a matrix
beginalign*
A = beginpmatrix
0 & -b_1 & 0 & -c_1 \
1 & -b_2 & 0 & -c_2 \
0 & -b_3 & 0 & -c_3 \
0 & -b_4 & 1 & -c_4 \
endpmatrix,
endalign*
then the charateristic polynomial is
beginalign*
t^4 + (c_4+b_2)t^3 + (c_3 + b_1 + b_2c_4 - b_4 c_3)t^2 + (b_1 c_4 - b_3 c_3 + b_2 c_3 - b_4 c_1) t + (b_1 c_3 - b_3 c_1).
endalign*
I am thinking we should be able to continuously decrease $b_3, b_4, c_1, c_2$ to $0$ while keeping the polynomial unchanged by changing $b_1, b_2, c_3, c_4$ accordingly.
Edit: Many thanks to user9527 for putting bounty on my question.
linear-algebra general-topology matrices polynomials connectedness
asked Aug 2 at 18:07
MyCindy2012
335
Let us consider a subset $S$ of $M_4(mathbb R)$ which has following form
beginalign*
beginpmatrix
0 & * & 0 & * \
1 & * & 0 & * \
0 & * & 0 & * \
0 & * & 1 & *
endpmatrix,
endalign*
where $*$ can assume any real number. It is also clear for any monic $4^th$ degree real polynomial, we can at least find one realization in $S$ since the upper left block and lower right block can be considered as in companion form. Let $f: S to mathbb R^n$ be the map sending the coefficients of characteristic polynomial to $mathbb R^n$.
My question is: suppose we have a square-free polynomial $p(t) = t^4 + a_3 t^3 + a_2 t^2 + a_1 t + a_0$, is $f^-1(a)$ a connected set in $M_n(mathbb R)$ where $a=(a_3, a_2, a_1, a_0)$? If we let $C$ denote the companion form of $p(t)$, which is an element of $S$, then $$f^-1(a) = V C V^-1: V in GL_4(mathbb R), V C V^-1 text is in above form .$$
I have a feeling we can first choose a realization in block form,
beginalign*
beginpmatrix
0 & * & 0 & 0 \
1 & * & 0 & 0 \
0 & 0 & 0 & * \
0 & 0 & 1 & *
endpmatrix,
endalign*
and then continuously change everything in $f^-1(a)$ into this form without changing the characteristic polynomial. Indeed, if we consider a matrix
beginalign*
A = beginpmatrix
0 & -b_1 & 0 & -c_1 \
1 & -b_2 & 0 & -c_2 \
0 & -b_3 & 0 & -c_3 \
0 & -b_4 & 1 & -c_4 \
endpmatrix,
endalign*
then the charateristic polynomial is
beginalign*
t^4 + (c_4+b_2)t^3 + (c_3 + b_1 + b_2c_4 - b_4 c_3)t^2 + (b_1 c_4 - b_3 c_3 + b_2 c_3 - b_4 c_1) t + (b_1 c_3 - b_3 c_1).
endalign*
I am thinking we should be able to continuously decrease $b_3, b_4, c_1, c_2$ to $0$ while keeping the polynomial unchanged by changing $b_1, b_2, c_3, c_4$ accordingly.
Edit: Many thanks to user9527 for putting bounty on my question.
linear-algebra general-topology matrices polynomials connectedness
asked Aug 2 at 18:07
MyCindy2012
335
edited yesterday
asked Aug 2 at 18:07
MyCindy2012
335
asked Aug 2 at 18:07
MyCindy2012
335
asked Aug 2 at 18:07
MyCindy2012
335
335
This question has an open bounty worth +100
reputation from user9527 ending ending at 2018-08-12 06:45:27Z">in 4 days.
This question has not received enough attention.
This question has an open bounty worth +100
reputation from user9527 ending ending at 2018-08-12 06:45:27Z">in 4 days.
This question has not received enough attention.
add a comment |Â
add a comment |Â
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