Gershgorin Circle theorem- implications
up vote
1
down vote
favorite
(I am considering only real matrices)
Does only hold that if the area of all Gershgorin Circles is positiv $Rightarrow$ the Matrix is positiv definit (trivial)
or does also follow the vice versa
the Matrix is positiv definit $Rightarrow$ the area of all Gershgorin Circles is positiv
matrices gershgorin-sets
asked Nov 20 at 9:59
baxbear
397
add a comment |
up vote
1
down vote
favorite
(I am considering only real matrices)
Does only hold that if the area of all Gershgorin Circles is positiv $Rightarrow$ the Matrix is positiv definit (trivial)
or does also follow the vice versa
the Matrix is positiv definit $Rightarrow$ the area of all Gershgorin Circles is positiv
matrices gershgorin-sets
asked Nov 20 at 9:59
baxbear
397
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
(I am considering only real matrices)
Does only hold that if the area of all Gershgorin Circles is positiv $Rightarrow$ the Matrix is positiv definit (trivial)
or does also follow the vice versa
the Matrix is positiv definit $Rightarrow$ the area of all Gershgorin Circles is positiv
matrices gershgorin-sets
asked Nov 20 at 9:59
baxbear
397
(I am considering only real matrices)
Does only hold that if the area of all Gershgorin Circles is positiv $Rightarrow$ the Matrix is positiv definit (trivial)
or does also follow the vice versa
the Matrix is positiv definit $Rightarrow$ the area of all Gershgorin Circles is positiv
matrices gershgorin-sets
matrices gershgorin-sets
asked Nov 20 at 9:59
baxbear
397
asked Nov 20 at 9:59
baxbear
397
asked Nov 20 at 9:59
baxbear
397
asked Nov 20 at 9:59
baxbear
397
asked Nov 20 at 9:59
baxbear
397
397
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
The reverse direction does not hold:
$$
A=pmatrix{ 1 & 2\ 2 & 10}
$$
is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.
answered Nov 20 at 10:13
daw
24k1544
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
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
up vote
2
down vote
accepted
The reverse direction does not hold:
$$
A=pmatrix{ 1 & 2\ 2 & 10}
$$
is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.
answered Nov 20 at 10:13
daw
24k1544
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
add a comment |
up vote
2
down vote
accepted
The reverse direction does not hold:
$$
A=pmatrix{ 1 & 2\ 2 & 10}
$$
is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.
answered Nov 20 at 10:13
daw
24k1544
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
The reverse direction does not hold:
$$
A=pmatrix{ 1 & 2\ 2 & 10}
$$
is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.
answered Nov 20 at 10:13
daw
24k1544
The reverse direction does not hold:
$$
A=pmatrix{ 1 & 2\ 2 & 10}
$$
is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.
answered Nov 20 at 10:13
daw
24k1544
answered Nov 20 at 10:13
daw
24k1544
answered Nov 20 at 10:13
daw
24k1544
answered Nov 20 at 10:13
daw
24k1544
24k1544
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
add a comment |
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
Thanks, I found something like it for special matrices in my old exercise papers and was wondering if I can use it to check if a $LL^T$ Cholesky decomposition is possible, seems it is not the case, thanks
– baxbear
Nov 20 at 10:17
add a comment |
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