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How many realizations are there then in a structured set of matrices yielding characteristic polynomial $(t+1)^4$?
Clash Royale CLAN TAG#URR8PPP
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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.
Is there a systemic way to determine how many realizations there are of $p(t) = (t+1)^4$ in $S$ up to Jordan forms? That is, I would like to know whether
all Jordan blocks with diagonals to be $-1$'s are realizable in $S$? Obviously,
beginalign*
J_1 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 0 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix, quad J_2 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 1 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix
endalign*
are realizable by choosing
beginalign*
A_1 = beginpmatrix
0 & -1 & 0 & 0 \
1 & -2 & 0 & 0 \
0 & 0 & 0 & -1\
0 & 0 & 1 & -2
endpmatrix, quad
A_2 = beginpmatrix
0 & 0 & 0 & -1 \
1 & 0 & 0 & -4 \
0 & 1 & 0 & -6\
0 & 0 & 1 & -4
endpmatrix,
endalign*
For other cases, I am essentially experimenting and try. But with no luck to get any result.
linear-algebra matrices jordan-normal-form
asked Aug 6 at 18:20
MyCindy2012
535
 |Â
show 1 more comment
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.
Is there a systemic way to determine how many realizations there are of $p(t) = (t+1)^4$ in $S$ up to Jordan forms? That is, I would like to know whether
all Jordan blocks with diagonals to be $-1$'s are realizable in $S$? Obviously,
beginalign*
J_1 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 0 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix, quad J_2 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 1 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix
endalign*
are realizable by choosing
beginalign*
A_1 = beginpmatrix
0 & -1 & 0 & 0 \
1 & -2 & 0 & 0 \
0 & 0 & 0 & -1\
0 & 0 & 1 & -2
endpmatrix, quad
A_2 = beginpmatrix
0 & 0 & 0 & -1 \
1 & 0 & 0 & -4 \
0 & 1 & 0 & -6\
0 & 0 & 1 & -4
endpmatrix,
endalign*
For other cases, I am essentially experimenting and try. But with no luck to get any result.
linear-algebra matrices jordan-normal-form
asked Aug 6 at 18:20
MyCindy2012
535
You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
1
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32
 |Â
show 1 more comment
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.
Is there a systemic way to determine how many realizations there are of $p(t) = (t+1)^4$ in $S$ up to Jordan forms? That is, I would like to know whether
all Jordan blocks with diagonals to be $-1$'s are realizable in $S$? Obviously,
beginalign*
J_1 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 0 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix, quad J_2 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 1 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix
endalign*
are realizable by choosing
beginalign*
A_1 = beginpmatrix
0 & -1 & 0 & 0 \
1 & -2 & 0 & 0 \
0 & 0 & 0 & -1\
0 & 0 & 1 & -2
endpmatrix, quad
A_2 = beginpmatrix
0 & 0 & 0 & -1 \
1 & 0 & 0 & -4 \
0 & 1 & 0 & -6\
0 & 0 & 1 & -4
endpmatrix,
endalign*
For other cases, I am essentially experimenting and try. But with no luck to get any result.
linear-algebra matrices jordan-normal-form
asked Aug 6 at 18:20
MyCindy2012
535
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.
Is there a systemic way to determine how many realizations there are of $p(t) = (t+1)^4$ in $S$ up to Jordan forms? That is, I would like to know whether
all Jordan blocks with diagonals to be $-1$'s are realizable in $S$? Obviously,
beginalign*
J_1 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 0 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix, quad J_2 = beginpmatrix
-1 & 0 & 0 & 0 \
1 & -1 & 0 & 0 \
0 & 1 & -1 & 0 \
0 & 0 & 1 & -1
endpmatrix
endalign*
are realizable by choosing
beginalign*
A_1 = beginpmatrix
0 & -1 & 0 & 0 \
1 & -2 & 0 & 0 \
0 & 0 & 0 & -1\
0 & 0 & 1 & -2
endpmatrix, quad
A_2 = beginpmatrix
0 & 0 & 0 & -1 \
1 & 0 & 0 & -4 \
0 & 1 & 0 & -6\
0 & 0 & 1 & -4
endpmatrix,
endalign*
For other cases, I am essentially experimenting and try. But with no luck to get any result.
linear-algebra matrices jordan-normal-form
asked Aug 6 at 18:20
MyCindy2012
535
edited Aug 6 at 18:59
asked Aug 6 at 18:20
MyCindy2012
535
asked Aug 6 at 18:20
MyCindy2012
535
asked Aug 6 at 18:20
MyCindy2012
535
535
You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
1
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32
 |Â
show 1 more comment
You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
1
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32
You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
1
1
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32
 |Â
show 1 more comment
1 Answer
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The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.
Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = textdiag(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M in mathbbR^4 times 4$ be a matrix of your desired shape and $M = A cdot J cdot A^-1$ with some matrix $A in GL_4(mathbbR)$. This similarity would imply:
$$ M = J_d + A N A^-1.$$
Since $N$ is of rank 1, so is $A N A^-1$, so it can be represented in the form
$$ANA^-1 = v cdot w^T$$
with vectors $v,w in mathbbR^4.$ Writing out the equality $M = J_d + v cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$2 = v_1 w_1, quad
0 = v_1 w_3, quad
2 = v_3 w_3.$$
From these follow $w_1 neq 0, w_3 neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = frac2w_1$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$beginpmatrix
0 &1 &0 &-2 \
1 &0 &0 &-2 \
0 &2 &0 &-3 \
0 &2 &1 &-4 \
endpmatrix,$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$tag*$square$$$
How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d = beginpmatrix
1 &* &0 &* \
1 &* &0 &* \
0 &* &1 &* \
0 &* &1 &* \
endpmatrix.$$
Note that the matrix $M - J_d = ANA^-1$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = beginpmatrix
1 &a &0 &c \
1 &a &0 &c \
0 &b &1 &d \
0 &b &1 &d \
endpmatrix$$
with parameters $a,b,c,d in mathbbR$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,, b = 2,, c = -2,, d = -3$, fulfilling the equation.
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.
Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = textdiag(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M in mathbbR^4 times 4$ be a matrix of your desired shape and $M = A cdot J cdot A^-1$ with some matrix $A in GL_4(mathbbR)$. This similarity would imply:
$$ M = J_d + A N A^-1.$$
Since $N$ is of rank 1, so is $A N A^-1$, so it can be represented in the form
$$ANA^-1 = v cdot w^T$$
with vectors $v,w in mathbbR^4.$ Writing out the equality $M = J_d + v cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$2 = v_1 w_1, quad
0 = v_1 w_3, quad
2 = v_3 w_3.$$
From these follow $w_1 neq 0, w_3 neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = frac2w_1$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$beginpmatrix
0 &1 &0 &-2 \
1 &0 &0 &-2 \
0 &2 &0 &-3 \
0 &2 &1 &-4 \
endpmatrix,$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$tag*$square$$$
How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d = beginpmatrix
1 &* &0 &* \
1 &* &0 &* \
0 &* &1 &* \
0 &* &1 &* \
endpmatrix.$$
Note that the matrix $M - J_d = ANA^-1$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = beginpmatrix
1 &a &0 &c \
1 &a &0 &c \
0 &b &1 &d \
0 &b &1 &d \
endpmatrix$$
with parameters $a,b,c,d in mathbbR$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,, b = 2,, c = -2,, d = -3$, fulfilling the equation.
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
add a comment |Â
up vote
1
down vote
accepted
The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.
Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = textdiag(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M in mathbbR^4 times 4$ be a matrix of your desired shape and $M = A cdot J cdot A^-1$ with some matrix $A in GL_4(mathbbR)$. This similarity would imply:
$$ M = J_d + A N A^-1.$$
Since $N$ is of rank 1, so is $A N A^-1$, so it can be represented in the form
$$ANA^-1 = v cdot w^T$$
with vectors $v,w in mathbbR^4.$ Writing out the equality $M = J_d + v cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$2 = v_1 w_1, quad
0 = v_1 w_3, quad
2 = v_3 w_3.$$
From these follow $w_1 neq 0, w_3 neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = frac2w_1$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$beginpmatrix
0 &1 &0 &-2 \
1 &0 &0 &-2 \
0 &2 &0 &-3 \
0 &2 &1 &-4 \
endpmatrix,$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$tag*$square$$$
How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d = beginpmatrix
1 &* &0 &* \
1 &* &0 &* \
0 &* &1 &* \
0 &* &1 &* \
endpmatrix.$$
Note that the matrix $M - J_d = ANA^-1$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = beginpmatrix
1 &a &0 &c \
1 &a &0 &c \
0 &b &1 &d \
0 &b &1 &d \
endpmatrix$$
with parameters $a,b,c,d in mathbbR$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,, b = 2,, c = -2,, d = -3$, fulfilling the equation.
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.
Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = textdiag(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M in mathbbR^4 times 4$ be a matrix of your desired shape and $M = A cdot J cdot A^-1$ with some matrix $A in GL_4(mathbbR)$. This similarity would imply:
$$ M = J_d + A N A^-1.$$
Since $N$ is of rank 1, so is $A N A^-1$, so it can be represented in the form
$$ANA^-1 = v cdot w^T$$
with vectors $v,w in mathbbR^4.$ Writing out the equality $M = J_d + v cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$2 = v_1 w_1, quad
0 = v_1 w_3, quad
2 = v_3 w_3.$$
From these follow $w_1 neq 0, w_3 neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = frac2w_1$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$beginpmatrix
0 &1 &0 &-2 \
1 &0 &0 &-2 \
0 &2 &0 &-3 \
0 &2 &1 &-4 \
endpmatrix,$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$tag*$square$$$
How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d = beginpmatrix
1 &* &0 &* \
1 &* &0 &* \
0 &* &1 &* \
0 &* &1 &* \
endpmatrix.$$
Note that the matrix $M - J_d = ANA^-1$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = beginpmatrix
1 &a &0 &c \
1 &a &0 &c \
0 &b &1 &d \
0 &b &1 &d \
endpmatrix$$
with parameters $a,b,c,d in mathbbR$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,, b = 2,, c = -2,, d = -3$, fulfilling the equation.
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
The Jordan form of a matrix of your desired shape with characteristic polynomial $(1 + t)^4$ has nilpotent part of rank 2 or higher, and all such Jordan forms can be realized by such a matrix.
Split up the Jordan form $J$ into a diagonal part $J_d$ and a lower triangular nilpotent part $N$:
$$J = J_d + N,$$
where $J_d = textdiag(-1,-1,-1,-1)$. This is super handy, because we have the special case where $J_d$ commutes with every matrix. I am going to divide this up into a few cases depending on the rank of the matrix.
Case 1: $N = 0$. Then, it is not possible for $J$ to be similar to a matrix of your desired form; the only matrix similar to $J = J_d$ is $J_d$ itself, which cannot be represented in the form you seek.
Case 2: $N$ is a rank 1 matrix. A such representation is in this case also not possible: Let $M in mathbbR^4 times 4$ be a matrix of your desired shape and $M = A cdot J cdot A^-1$ with some matrix $A in GL_4(mathbbR)$. This similarity would imply:
$$ M = J_d + A N A^-1.$$
Since $N$ is of rank 1, so is $A N A^-1$, so it can be represented in the form
$$ANA^-1 = v cdot w^T$$
with vectors $v,w in mathbbR^4.$ Writing out the equality $M = J_d + v cdot w^T$ yields an incompatible set of equations in the first and third column: Taking the equation for the $(1,1)$-,$(3,1)$- and $(3,3)$-entries for example yields
$$2 = v_1 w_1, quad
0 = v_1 w_3, quad
2 = v_3 w_3.$$
From these follow $w_1 neq 0, w_3 neq 0$, and thus simultaneously $v_1 = 0$ and $v_1 = frac2w_1$.
Case 3: $N$ is of rank 2 with one Jordan block of size 3 and one of size 1. For this case, consider for example the matrix
$$beginpmatrix
0 &1 &0 &-2 \
1 &0 &0 &-2 \
0 &2 &0 &-3 \
0 &2 &1 &-4 \
endpmatrix,$$
which is in your desired form and similar to a matrix with one Jordan block of size 3 and one of size 1. I give an explanation for how I found this example below. All other cases were already handled in the OP. $$tag*$square$$$
How I derived the last example: The case where $N$ possesses Jordan blocks of size 1 and 3 is equivalent to demanding $(M-J_d)^2 neq 0$ but $(M-J_d)^3 = 0$.
Remember:
$$M - J_d = beginpmatrix
1 &* &0 &* \
1 &* &0 &* \
0 &* &1 &* \
0 &* &1 &* \
endpmatrix.$$
Note that the matrix $M - J_d = ANA^-1$ is of rank 2, as $N$ is of rank 2, and $M - J_d$ already possesses two linearly independent vectors in column 1 and 3, so column 2 and 4 need to be linear combinations of those. As such, the most general form for $M - J_d$ is
$$M - J_d = beginpmatrix
1 &a &0 &c \
1 &a &0 &c \
0 &b &1 &d \
0 &b &1 &d \
endpmatrix$$
with parameters $a,b,c,d in mathbbR$. By writing the equations down explicitly, one easily receives that $(M - J_d)^2 = 0$ is equivalent to the conditions $a = d = -1, b = c = 0$.
The equations for $(M - J_d)^3 = 0$ are a fair bit more convoluted (click here), but you can play around with these entries a bit, and for example demand that $d = -a - 2$, from which you then simply get the easy condition
$$(a + 1)^2 + bc = 0.$$
If this is fulfilled, $(M - J_d)^3$ vanishes, and now you just need to pick the parameters in a way that $(M - J_d)^2$ does not vanish. In the matrix example I gave above, I, for example, picked the parameters $a = 1,, b = 2,, c = -2,, d = -3$, fulfilling the equation.
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
edited Aug 7 at 12:26
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
answered Aug 6 at 19:50
Lukas Miristwhisky
330111
330111
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You mean Jordan blocks with diagonals to be all $-1$, right?
– amarney
Aug 6 at 18:54
Yes. Thanks for spotting the typo.
– MyCindy2012
Aug 6 at 18:59
1
My guess is to look at geometric multiplicities of your eigenvalues, since they are what determines the size and number of your Jordan blocks. Maybe look at the equation beginalign (A - lambda I)x = beginbmatrix -lambda & * & 0 & * \ 1 & *-lambda & 0 & * \ 0 & * & - lambda & * \ 0 & * & 1 & *-lambda endbmatrixx =0. endalign
– amarney
Aug 6 at 19:06
I edited my partial answer and it is now a full one. Using that the diagonal part of the Jordan form commutes with everything is a very handy feature here, and it simplifies the calculations so that they can be carried out explicitly :)
– Lukas Miristwhisky
Aug 6 at 20:45
Update: I noticed my "Case 3" was wrong, because I assumed that a nilpotent matrix of rank 2 has $N^2 = 0$. This is wrong, as can be seen in the case of a Jordan block of size 3. Instead, I found an example for this case. It should now be complete and correct. Sorry!
– Lukas Miristwhisky
Aug 7 at 12:32