Mixing
Sparsest similar matrix
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser than other if it has less nonzero entries.
Two square $n times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^-1CP$
linear-algebra matrices sparse-matrices
asked 2 hours ago
Nacho Garcia Marco
263
add a comment |Â
up vote
4
down vote
favorite
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser than other if it has less nonzero entries.
Two square $n times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^-1CP$
linear-algebra matrices sparse-matrices
asked 2 hours ago
Nacho Garcia Marco
263
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
2
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser than other if it has less nonzero entries.
Two square $n times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^-1CP$
linear-algebra matrices sparse-matrices
asked 2 hours ago
Nacho Garcia Marco
263
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser than other if it has less nonzero entries.
Two square $n times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^-1CP$
linear-algebra matrices sparse-matrices
linear-algebra matrices sparse-matrices
asked 2 hours ago
Nacho Garcia Marco
263
asked 2 hours ago
Nacho Garcia Marco
263
asked 2 hours ago
Nacho Garcia Marco
263
asked 2 hours ago
Nacho Garcia Marco
263
asked 2 hours ago
Nacho Garcia Marco
263
263
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
2
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago
add a comment |Â
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
2
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
2
2
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$pmatrix0&0&0&-1cr1&0&0&0cr0&1&0&2cr0&0&1&0cr$$ which has Jordan form $$pmatrix1&1&0&0cr0&1&0&0cr0&0&-1&1cr0&0&0&-1cr$$ which has more nonzero entries.
answered 1 hour ago
Gerry Myerson
145k8145296
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$pmatrix0&0&0&-1cr1&0&0&0cr0&1&0&2cr0&0&1&0cr$$ which has Jordan form $$pmatrix1&1&0&0cr0&1&0&0cr0&0&-1&1cr0&0&0&-1cr$$ which has more nonzero entries.
answered 1 hour ago
Gerry Myerson
145k8145296
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
add a comment |Â
up vote
5
down vote
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$pmatrix0&0&0&-1cr1&0&0&0cr0&1&0&2cr0&0&1&0cr$$ which has Jordan form $$pmatrix1&1&0&0cr0&1&0&0cr0&0&-1&1cr0&0&0&-1cr$$ which has more nonzero entries.
answered 1 hour ago
Gerry Myerson
145k8145296
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
add a comment |Â
up vote
5
down vote
up vote
5
down vote
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$pmatrix0&0&0&-1cr1&0&0&0cr0&1&0&2cr0&0&1&0cr$$ which has Jordan form $$pmatrix1&1&0&0cr0&1&0&0cr0&0&-1&1cr0&0&0&-1cr$$ which has more nonzero entries.
answered 1 hour ago
Gerry Myerson
145k8145296
The companion matrix to the polynomial $(x^2-1)^2=1-2x^2+x^4$ is $$pmatrix0&0&0&-1cr1&0&0&0cr0&1&0&2cr0&0&1&0cr$$ which has Jordan form $$pmatrix1&1&0&0cr0&1&0&0cr0&0&-1&1cr0&0&0&-1cr$$ which has more nonzero entries.
answered 1 hour ago
Gerry Myerson
145k8145296
answered 1 hour ago
Gerry Myerson
145k8145296
answered 1 hour ago
Gerry Myerson
145k8145296
answered 1 hour ago
Gerry Myerson
145k8145296
145k8145296
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
add a comment |Â
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
The following paper claims that the companion matrix is the sparsest in a certain sense: doi.org/10.1016/j.laa.2012.08.017
– Julian Kuelshammer
34 mins ago
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2961899%2fsparsest-similar-matrix%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
If the matrix is diagonalisable, clearly its Jordan form is the sparsest one in its similarity class because you cannot get fewer than $operatornamerank(A)$ entries.
– user1551
2 hours ago
My intuition says the same, but apart from trivial cases ($A$ is diagonaliseable) I do not see an immediate proof of that.
– TZakrevskiy
2 hours ago
2
Simulposted to MO, mathoverflow.net/questions/313224/sparsest-similar-matrix DON'T DO THAT!
– Gerry Myerson
1 hour ago