Given any matrix $A$, does there exist a symplectic transformation such that $P^TAP=B$ where B is block...
$begingroup$
Given any $2ntimes 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where
$$
B=begin{bmatrix}
B_1&0&cdots&0\
0&B_2&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&B_n
end{bmatrix}
$$
and
$$
B_i=begin{bmatrix}
0&lambda_i\-lambda_i&0
end{bmatrix}
$$
By
saying that $P$ is matrix associated with above symplectic transformation, we mean that $P$ satisfy the condition of $P^TJP=J$, where
$$
J=begin{bmatrix}
J_0&0&cdots&0\
0&J_0&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&J_0
end{bmatrix}
$$
and
$$
J_0=begin{bmatrix}
0&1\-1&0
end{bmatrix}
$$
linear-algebra diagonalization symplectic-geometry block-matrices
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
$endgroup$
add a comment |
$begingroup$
Given any $2ntimes 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where
$$
B=begin{bmatrix}
B_1&0&cdots&0\
0&B_2&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&B_n
end{bmatrix}
$$
and
$$
B_i=begin{bmatrix}
0&lambda_i\-lambda_i&0
end{bmatrix}
$$
By
saying that $P$ is matrix associated with above symplectic transformation, we mean that $P$ satisfy the condition of $P^TJP=J$, where
$$
J=begin{bmatrix}
J_0&0&cdots&0\
0&J_0&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&J_0
end{bmatrix}
$$
and
$$
J_0=begin{bmatrix}
0&1\-1&0
end{bmatrix}
$$
linear-algebra diagonalization symplectic-geometry block-matrices
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
$endgroup$
add a comment |
$begingroup$
Given any $2ntimes 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where
$$
B=begin{bmatrix}
B_1&0&cdots&0\
0&B_2&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&B_n
end{bmatrix}
$$
and
$$
B_i=begin{bmatrix}
0&lambda_i\-lambda_i&0
end{bmatrix}
$$
By
saying that $P$ is matrix associated with above symplectic transformation, we mean that $P$ satisfy the condition of $P^TJP=J$, where
$$
J=begin{bmatrix}
J_0&0&cdots&0\
0&J_0&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&J_0
end{bmatrix}
$$
and
$$
J_0=begin{bmatrix}
0&1\-1&0
end{bmatrix}
$$
linear-algebra diagonalization symplectic-geometry block-matrices
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
$endgroup$
Given any $2ntimes 2n$ matrix $A$, does there always exist a symplectic transformation such that $P^TAP=B$ where B is block diagonal? where
$$
B=begin{bmatrix}
B_1&0&cdots&0\
0&B_2&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&B_n
end{bmatrix}
$$
and
$$
B_i=begin{bmatrix}
0&lambda_i\-lambda_i&0
end{bmatrix}
$$
By
saying that $P$ is matrix associated with above symplectic transformation, we mean that $P$ satisfy the condition of $P^TJP=J$, where
$$
J=begin{bmatrix}
J_0&0&cdots&0\
0&J_0&cdots&0\
0&cdots&ddots&vdots\
0&0&cdots&J_0
end{bmatrix}
$$
and
$$
J_0=begin{bmatrix}
0&1\-1&0
end{bmatrix}
$$
linear-algebra diagonalization symplectic-geometry block-matrices
linear-algebra diagonalization symplectic-geometry block-matrices
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
asked Nov 27 '18 at 23:42
MclalalalaMclalalala
1269
1269
add a comment |
add a comment |
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