Symmetric and definite positive matrix - complex vs real coefficients
Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite.
Is it true that
$$A := {^t M} N^{-1} overline M $$
has real coefficients?
(Here $^t M=M$ by symmetry).
I tested a random example, it seems to work. But writing the $ij$-coefficient of $A$ gives
$$sum_{k,l} (^t M)_{il} (N^{-1})_{lk} overline M_{kj}$$
so I'm not sure what to do.
linear-algebra matrices positive-definite
asked Nov 20 at 11:10
Alphonse
2,178623
add a comment |
Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite.
Is it true that
$$A := {^t M} N^{-1} overline M $$
has real coefficients?
(Here $^t M=M$ by symmetry).
I tested a random example, it seems to work. But writing the $ij$-coefficient of $A$ gives
$$sum_{k,l} (^t M)_{il} (N^{-1})_{lk} overline M_{kj}$$
so I'm not sure what to do.
linear-algebra matrices positive-definite
asked Nov 20 at 11:10
Alphonse
2,178623
add a comment |
Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite.
Is it true that
$$A := {^t M} N^{-1} overline M $$
has real coefficients?
(Here $^t M=M$ by symmetry).
I tested a random example, it seems to work. But writing the $ij$-coefficient of $A$ gives
$$sum_{k,l} (^t M)_{il} (N^{-1})_{lk} overline M_{kj}$$
so I'm not sure what to do.
linear-algebra matrices positive-definite
asked Nov 20 at 11:10
Alphonse
2,178623
Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite.
Is it true that
$$A := {^t M} N^{-1} overline M $$
has real coefficients?
(Here $^t M=M$ by symmetry).
I tested a random example, it seems to work. But writing the $ij$-coefficient of $A$ gives
$$sum_{k,l} (^t M)_{il} (N^{-1})_{lk} overline M_{kj}$$
so I'm not sure what to do.
linear-algebra matrices positive-definite
linear-algebra matrices positive-definite
asked Nov 20 at 11:10
Alphonse
2,178623
asked Nov 20 at 11:10
Alphonse
2,178623
edited Nov 20 at 11:24
asked Nov 20 at 11:10
Alphonse
2,178623
asked Nov 20 at 11:10
Alphonse
2,178623
asked Nov 20 at 11:10
Alphonse
2,178623
2,178623
add a comment |
add a comment |
1 Answer
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Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then
$$
A=M^TN^{-1}overline{M}=MN^{-1}overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N,
$$
which is real.
answered Nov 20 at 11:38
user1551
71.2k566125
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then
$$
A=M^TN^{-1}overline{M}=MN^{-1}overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N,
$$
which is real.
answered Nov 20 at 11:38
user1551
71.2k566125
add a comment |
Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then
$$
A=M^TN^{-1}overline{M}=MN^{-1}overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N,
$$
which is real.
answered Nov 20 at 11:38
user1551
71.2k566125
add a comment |
Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then
$$
A=M^TN^{-1}overline{M}=MN^{-1}overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N,
$$
which is real.
answered Nov 20 at 11:38
user1551
71.2k566125
Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then
$$
A=M^TN^{-1}overline{M}=MN^{-1}overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N,
$$
which is real.
answered Nov 20 at 11:38
user1551
71.2k566125
answered Nov 20 at 11:38
user1551
71.2k566125
answered Nov 20 at 11:38
user1551
71.2k566125
answered Nov 20 at 11:38
user1551
71.2k566125
71.2k566125
add a comment |
add a comment |
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