Positive semidefinite cone - Revision history

Dattorro at 05:56, 9 July 2011

Dattorro — Sat, 09 Jul 2011 05:56:26 GMT

←Older revision		Revision as of 05:56, 9 July 2011
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	The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.

-	In low dimension, the positive semidefinite cone ~~can be~~ shown to be a circular cone by way of an isometric isomorphism <math>T</math> relating matrix space to vector space:	+	In low dimension, the positive semidefinite cone is shown to be a circular cone by way of an isometric isomorphism <math>T</math> relating matrix space to vector space:
	<ul>		<ul>
	<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>		<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>

Dattorro at 05:54, 9 July 2011

Dattorro — Sat, 09 Jul 2011 05:54:50 GMT

←Older revision		Revision as of 05:54, 9 July 2011
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	The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.

-	In low dimension, the positive semidefinite cone is a circular cone ~~because there is~~ an isometric isomorphism <math>T</math> relating matrix space to vector space:	+	In low dimension, the positive semidefinite cone can be shown to be a circular cone by way of an isometric isomorphism <math>T</math> relating matrix space to vector space:
	<ul>		<ul>
	<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>		<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>

Dattorro at 05:15, 9 July 2011

Dattorro — Sat, 09 Jul 2011 05:15:44 GMT

←Older revision		Revision as of 05:15, 9 July 2011
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	<ul>		<ul>
	<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>		<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>
-	<strong>(</strong>distance between matrices ~~equals~~ distance between vectorized matrices<strong>)</strong>.	+	<strong>(</strong>distance between matrices <math>=</math> distance between vectorized matrices<strong>)</strong>.
	<li>In one dimension, 1×1 symmetric matrices, the nonnegative ray is a circular cone.		<li>In one dimension, 1×1 symmetric matrices, the nonnegative ray is a circular cone.
	</ul>		</ul>

	[http://meboo.convexoptimization.com/access.html Read more...]		[http://meboo.convexoptimization.com/access.html Read more...]

Dattorro at 07:23, 8 July 2011

Dattorro — Fri, 08 Jul 2011 07:23:10 GMT

←Older revision		Revision as of 07:23, 8 July 2011
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	The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.

-	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space:	+	In low dimension, the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space:
	<ul>		<ul>
	<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>		<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>

Dattorro at 07:22, 8 July 2011

Dattorro — Fri, 08 Jul 2011 07:22:28 GMT

←Older revision		Revision as of 07:22, 8 July 2011
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	The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.

-	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>	+	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space:
		+	<ul>
		+	<li>For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>
	<strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>.		<strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>.
-	In one dimension, the nonnegative ray is a circular cone.	+	<li>In one dimension, 1×1 symmetric matrices, the nonnegative ray is a circular cone.
		+	</ul>

	[http://meboo.convexoptimization.com/access.html Read more...]		[http://meboo.convexoptimization.com/access.html Read more...]

Dattorro at 07:17, 8 July 2011

Dattorro — Fri, 08 Jul 2011 07:17:36 GMT

←Older revision		Revision as of 07:17, 8 July 2011
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	each halfspace having partial boundary containing the origin in an isomorphic subspace.		each halfspace having partial boundary containing the origin in an isomorphic subspace.

-	Hence the positive semidefinite cone~~ ~~is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.	+	Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.

	The positive definite <strong>(</strong>full-rank<strong>)</strong> matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one <math>0</math> eigenvalue<strong>)</strong> reside on the cone boundary.		The positive definite <strong>(</strong>full-rank<strong>)</strong> matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one <math>0</math> eigenvalue<strong>)</strong> reside on the cone boundary.

-	The only symmetric positive semidefinite matrix having~~ ~~all ~~zero~~ eigenvalues resides at the origin.	+	The only symmetric positive semidefinite matrix having all <math>0</math> eigenvalues resides at the origin.

	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>		In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>

Dattorro at 05:17, 8 July 2011

Dattorro — Fri, 08 Jul 2011 05:17:30 GMT

←Older revision		Revision as of 05:17, 8 July 2011
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	[[Image:psdcone.jpg\|right\|positive semidefinite cone is a circular cone in 3D]]		[[Image:psdcone.jpg\|right\|positive semidefinite cone is a circular cone in 3D]]
-	<br>
	The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone:		The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone:

-	It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix ~~from the~~ figure,	+	It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix
		+	<strong>(</strong>as in figure<strong>)</strong>,
	each halfspace having partial boundary containing the origin in an isomorphic subspace.		each halfspace having partial boundary containing the origin in an isomorphic subspace.

Dattorro at 05:05, 8 July 2011

Dattorro — Fri, 08 Jul 2011 05:05:54 GMT

←Older revision		Revision as of 05:05, 8 July 2011
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	Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.		Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.

-	The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one <math>0</math> eigenvalue<strong>)</strong> reside on the cone boundary.	+	The positive definite <strong>(</strong>full-rank<strong>)</strong> matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one <math>0</math> eigenvalue<strong>)</strong> reside on the cone boundary.

	The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.

-	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 (as in figure). This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>	+	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 <strong>(</strong>as in figure<strong>)</strong>. This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>
	<strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>.		<strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>.
	In one dimension, the nonnegative ray is a circular cone.		In one dimension, the nonnegative ray is a circular cone.

-	[http://meboo.convexoptimization.com/access.html Read more]...	+	[http://meboo.convexoptimization.com/access.html Read more...]

Dattorro at 05:01, 8 July 2011

Dattorro — Fri, 08 Jul 2011 05:01:33 GMT

←Older revision		Revision as of 05:01, 8 July 2011
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	[[Image:psdcone.jpg\|right\|positive semidefinite cone is a circular cone in 3D]]		[[Image:psdcone.jpg\|right\|positive semidefinite cone is a circular cone in 3D]]
-	<br />	+	<br>
-	The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone: ~~<br />~~	+	The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone:
-	It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix from the figure, ~~<br />~~	+
-	each halfspace having partial boundary containing the origin in an isomorphic subspace. Hence~~ ~~the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.	+	It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix from the figure,
-	~~<br>~~The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one 0 eigenvalue<strong>)</strong> reside on the cone boundary.~~<br />~~	+	each halfspace having partial boundary containing the origin in an isomorphic subspace.
-	~~<br />~~	+
		+	Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.
		+
		+	The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices <strong>(</strong>having at least one <math>0</math> eigenvalue<strong>)</strong> reside on the cone boundary.
		+
	The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.		The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.
-	~~<br>~~In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism T relating matrix space to vector space: For a 2×2 symmetric matrix, T is obtained by scaling the ß coordinate by v2 (as in figure). This linear bijective transformation T preserves distance between two points in each respective space; <i>i.e.,</i> \|\|x - y\|\|<sub>F</sub> = \|\|Tx - Ty\|\|<sub>2</sub> <strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>. In one dimension, the nonnegative ray is a circular cone.	+
-	~~<br>~~[http://meboo.convexoptimization.com/access.html Read more]...	+	In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism <math>T</math> relating matrix space to vector space: For a 2×2 symmetric matrix, <math>T</math> is obtained by scaling the ß coordinate by √2 (as in figure). This linear bijective transformation <math>T</math> preserves distance between two points in each respective space; <i>i.e.,</i> \|\|<math>x - y</math>\|\|<sub>F</sub> = \|\|<math>Tx - Ty</math>\|\|<sub>2</sub>
		+	<strong>(</strong>distance between matrices equals distance between vectorized matrices<strong>)</strong>.
		+	In one dimension, the nonnegative ray is a circular cone.
		+
		+	[http://meboo.convexoptimization.com/access.html Read more]...

Dattorro: New page: ''"The cone of positive semidefinite matrices studied in this section is arguably the most important of all non-polyhedral cones whose facial structure we completely understand."'' <...

Dattorro — Fri, 08 Jul 2011 04:51:43 GMT

New page: ''"The cone of positive semidefinite matrices studied in this section is arguably the most important of all non-polyhedral cones whose facial structure we completely understand."''  <...

New page

''"The cone of positive semidefinite matrices studied in this section is arguably the most important of all non-polyhedral cones whose facial structure we completely understand."''  <math>-</math>Alexander Barvinok

[[Image:psdcone.jpg|right|positive semidefinite cone is a circular cone in 3D]]
 
The set of all symmetric positive semidefinite matrices of particular dimension is called the positive semidefinite cone: 
It can be formed by intersection of an infinite number of halfspaces in the vectorized variable matrix from the figure, 
each halfspace having partial boundary containing the origin in an isomorphic subspace. Hence the positive semidefinite cone is convex. It is a unique immutable proper cone in the ambient space of symmetric matrices.
 The positive definite (full-rank) matrices comprise the cone interior, while all singular positive semidefinite matrices (having at least one 0 eigenvalue) reside on the cone boundary. 
 
The only symmetric positive semidefinite matrix having all zero eigenvalues resides at the origin.
 In low dimension the positive semidefinite cone is a circular cone because there is an isometric isomorphism T relating matrix space to vector space: For a 2×2 symmetric matrix, T is obtained by scaling the ß coordinate by v2 (as in figure). This linear bijective transformation T preserves distance between two points in each respective space; i.e., ||x - y||F = ||Tx - Ty||2 (distance between matrices equals distance between vectorized matrices). In one dimension, the nonnegative ray is a circular cone.
 [http://meboo.convexoptimization.com/access.html Read more]...