Maximal eigenvalue is a convex function. Why?
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Let $A$ be a symmetric real matrix. Let $f(A)=lambda_max(A)$ be its largest eigenvalue. Why is $f$ convex?
matrices eigenvalues-eigenvectors convex-analysis
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
asked Jun 17 '15 at 17:39
Polaris
386
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up vote
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Let $A$ be a symmetric real matrix. Let $f(A)=lambda_max(A)$ be its largest eigenvalue. Why is $f$ convex?
matrices eigenvalues-eigenvectors convex-analysis
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
asked Jun 17 '15 at 17:39
Polaris
386
What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
2
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $A$ be a symmetric real matrix. Let $f(A)=lambda_max(A)$ be its largest eigenvalue. Why is $f$ convex?
matrices eigenvalues-eigenvectors convex-analysis
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
asked Jun 17 '15 at 17:39
Polaris
386
Let $A$ be a symmetric real matrix. Let $f(A)=lambda_max(A)$ be its largest eigenvalue. Why is $f$ convex?
matrices eigenvalues-eigenvectors convex-analysis
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
asked Jun 17 '15 at 17:39
Polaris
386
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
edited Aug 22 at 5:37
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Jun 17 '15 at 17:39
Polaris
386
asked Jun 17 '15 at 17:39
Polaris
386
asked Jun 17 '15 at 17:39
Polaris
386
386
What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
2
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07
add a comment |Â
What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
2
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07
What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
2
2
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07
add a comment |Â
3 Answers
3
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2
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HINT:
$f(A) I - Asucceq 0$, $f(B) I - B succeq 0$ implies $(lambda f(A) + (1- lambda) f(B))I - ( lambda A + (1-lambda) B) succeq 0$
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
add a comment |Â
up vote
1
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we have that $lambda_max(A)=textmint: tgeq h, hin S(A)$ where $S(A)=h:hgeq langle A, xx^Trangle text , forall x in R^n text and x^Tx=1$.
The set $S(A)$ is convex. This is a convex function since min is a convex function on a convex set. So the function is convex.
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
add a comment |Â
up vote
1
down vote
$f(A) = sup_x x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A mapsto x^TAx$ is linear, hence convex.)
answered Aug 22 at 5:55
littleO
26.3k540102
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
HINT:
$f(A) I - Asucceq 0$, $f(B) I - B succeq 0$ implies $(lambda f(A) + (1- lambda) f(B))I - ( lambda A + (1-lambda) B) succeq 0$
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
add a comment |Â
up vote
2
down vote
HINT:
$f(A) I - Asucceq 0$, $f(B) I - B succeq 0$ implies $(lambda f(A) + (1- lambda) f(B))I - ( lambda A + (1-lambda) B) succeq 0$
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
add a comment |Â
up vote
2
down vote
up vote
2
down vote
HINT:
$f(A) I - Asucceq 0$, $f(B) I - B succeq 0$ implies $(lambda f(A) + (1- lambda) f(B))I - ( lambda A + (1-lambda) B) succeq 0$
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
HINT:
$f(A) I - Asucceq 0$, $f(B) I - B succeq 0$ implies $(lambda f(A) + (1- lambda) f(B))I - ( lambda A + (1-lambda) B) succeq 0$
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
answered Jun 18 '15 at 7:05
orangeskid
28.1k31746
28.1k31746
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
add a comment |Â
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
Just Courant-Fischer in disguise
– orangeskid
Jun 18 '15 at 7:10
add a comment |Â
up vote
1
down vote
we have that $lambda_max(A)=textmint: tgeq h, hin S(A)$ where $S(A)=h:hgeq langle A, xx^Trangle text , forall x in R^n text and x^Tx=1$.
The set $S(A)$ is convex. This is a convex function since min is a convex function on a convex set. So the function is convex.
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
add a comment |Â
up vote
1
down vote
we have that $lambda_max(A)=textmint: tgeq h, hin S(A)$ where $S(A)=h:hgeq langle A, xx^Trangle text , forall x in R^n text and x^Tx=1$.
The set $S(A)$ is convex. This is a convex function since min is a convex function on a convex set. So the function is convex.
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
add a comment |Â
up vote
1
down vote
up vote
1
down vote
we have that $lambda_max(A)=textmint: tgeq h, hin S(A)$ where $S(A)=h:hgeq langle A, xx^Trangle text , forall x in R^n text and x^Tx=1$.
The set $S(A)$ is convex. This is a convex function since min is a convex function on a convex set. So the function is convex.
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
we have that $lambda_max(A)=textmint: tgeq h, hin S(A)$ where $S(A)=h:hgeq langle A, xx^Trangle text , forall x in R^n text and x^Tx=1$.
The set $S(A)$ is convex. This is a convex function since min is a convex function on a convex set. So the function is convex.
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
edited Aug 22 at 5:25
Laurent Lessard
1,4231212
1,4231212
answered Jun 17 '15 at 17:55
user85361
341215
answered Jun 17 '15 at 17:55
user85361
341215
answered Jun 17 '15 at 17:55
user85361
341215
341215
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
add a comment |Â
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
Thank you. I edited the answer.
– user85361
Jun 17 '15 at 19:47
add a comment |Â
up vote
1
down vote
$f(A) = sup_x x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A mapsto x^TAx$ is linear, hence convex.)
answered Aug 22 at 5:55
littleO
26.3k540102
add a comment |Â
up vote
1
down vote
$f(A) = sup_x x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A mapsto x^TAx$ is linear, hence convex.)
answered Aug 22 at 5:55
littleO
26.3k540102
add a comment |Â
up vote
1
down vote
up vote
1
down vote
$f(A) = sup_x x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A mapsto x^TAx$ is linear, hence convex.)
answered Aug 22 at 5:55
littleO
26.3k540102
$f(A) = sup_x x^T Ax$, and a supremum of convex functions is convex.
(For a given $x$, the function $A mapsto x^TAx$ is linear, hence convex.)
answered Aug 22 at 5:55
littleO
26.3k540102
edited Aug 26 at 17:04
answered Aug 22 at 5:55
littleO
26.3k540102
answered Aug 22 at 5:55
littleO
26.3k540102
answered Aug 22 at 5:55
littleO
26.3k540102
26.3k540102
add a comment |Â
add a comment |Â
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What is your definition of convex function?
– Matthew Leingang
Jun 17 '15 at 17:41
2
This follows from the min-max theorem
– Omnomnomnom
Jun 17 '15 at 17:44
More generally one has the Weyl inequalities, see en.wikipedia.org/wiki/…. And then, there is the Horn problem ...
– orangeskid
Jun 18 '15 at 7:07