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A positive definite matrix, by definition, has the property that all [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are positive. Beyond this:
 * A positive definite matrix is also [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric]].
 * All pivots are positive.
 * The [[LinearAlgebra/Determinant|determinant]] is positive, and all subdeterminants are also positive.		 A positive definite matrix, by definition, is a [[LinearAlgebra/SpecialMatrices#Symmetric_Matrices|symmetric]] matrix whose [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are all positive.
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'Semi-definite' is a slight modification, allowing 0. Such a matrix has several useful properties:
 * Always [[LinearAlgebra/Invertibility|invertible]]
 * All pivots are positive
 * The [[LinearAlgebra/Determinant|determinant]] is positive, and all subdeterminants are also positive

'Semi-definite' is a slight modification, allowing 0 as an eigenvalue.



=== Operations ===

If '''''A''''' is positive definite, then so is ''c'''A''''' for any real scalar ''c''.

If '''''A''''' and '''''B''''' are both positive definite...
 * so is '''''A''' + '''B'''''.
 * so is '''''ABA'''''.
 * so is '''''BAB'''''.

If '''''A''''' and '''''B''''' are both positive semi-definite, then so is '''''A''' + '''B'''''.

If '''''A''''' is definite and '''''B''''' is semi-definite, then '''''A''' + '''B''''' is definite.

Positive Definiteness

A matrix is positive definite if all eigenvalues are positive. A matrix is positive semi-definite if all eigenvalues are positive or zero.

Contents

  1. Positive Definiteness
    1. Description
      1. Operations

Description

A positive definite matrix, by definition, is a symmetric matrix whose eigenvalues are all positive.

Such a matrix has several useful properties:

  • Always invertible

  • All pivots are positive

  • The determinant is positive, and all subdeterminants are also positive

'Semi-definite' is a slight modification, allowing 0 as an eigenvalue.

Operations

If A is positive definite, then so is cA for any real scalar c.

If A and B are both positive definite...

  • so is A + B.

  • so is ABA.

  • so is BAB.

If A and B are both positive semi-definite, then so is A + B.

If A is definite and B is semi-definite, then A + B is definite.

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LinearAlgebra/PositiveDefiniteness (last edited 2026-02-02 05:34:32 by DominicRicottone)