Positive Definite Matrices #
This file defines positive (semi)definite matrices and connects the notion to positive definiteness of quadratic forms.
Main definition #
Matrix.PosDef
: a matrixM : Matrix n n 𝕜
is positive definite if it is hermitian andxᴴMx
is greater than zero for all nonzerox
.Matrix.PosSemidef
: a matrixM : Matrix n n 𝕜
is positive semidefinite if it is hermitian andxᴴMx
is nonnegative for allx
.
A matrix M : Matrix n n R
is positive definite if it is hermitian
and xᴴMx
is greater than zero for all nonzero x
.
Instances For
A matrix M : Matrix n n R
is positive semidefinite if it is hermitian
and xᴴMx
is nonnegative for all x
.
Instances For
The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite
A matrix multiplied by its conjugate transpose is positive semidefinite
The eigenvalues of a positive definite matrix are positive
The eigenvalues of a positive semi-definite matrix are non-negative
A positive definite matrix M
induces a norm ‖x‖ = sqrt (re xᴴMx)
.
Instances For
A positive definite matrix M
induces an inner product ⟪x, y⟫ = xᴴMy
.