Matrix sign definitions

An n×nn\times n matrix A\mathrm{A} takes on one of the following definitions if, for any vector v∈Rn\mathbf{v}\in \mathbb{R}^{n}, the following holds:

  • it is positive definite if v⋅Av=vTAv>0\mathbf{v}\cdot \mathrm{A}\mathbf{v}=\mathbf{v}^{T}\mathrm{A}\mathbf{v}>0
  • it is positive semidefinite if v⋅Av=vTAv≥0\mathbf{v}\cdot \mathrm{A}\mathbf{v}=\mathbf{v}^{T}\mathrm{A}\mathbf{v}\geq0
  • it is negative semidefinite if v⋅Av=vTAv≤0\mathbf{v}\cdot \mathrm{A}\mathbf{v}=\mathbf{v}^{T}\mathrm{A}\mathbf{v}\leq0
  • it is negative definite if v⋅Av=vTAv<0\mathbf{v}\cdot \mathrm{A}\mathbf{v}=\mathbf{v}^{T}\mathrm{A}\mathbf{v}<0 where T^{T} denotes transposition.

It's possible to make the same definitions by looking at the eigenvalues λi\lambda_{i} of A\mathrm{A}:

  • it is positive definite if λi>0\lambda_{i}>0
  • it is positive semidefinite if λi≥0\lambda_{i}\geq0
  • it is negative semidefinite if λi≤0\lambda_{i}\leq0
  • it is negative definite if λi<0\lambda_{i}<0 for all i=1,…,ni=1,\ldots,n.

Another set of equivalent definitions uses the determinant of the matrix:

  • it is positive definite if det⁡A>0\det \mathrm{A}>0
  • it is positive semidefinite if det⁡A≥0\det \mathrm{A}\geq0
  • it is negative semidefinite if det⁡A≤0\det \mathrm{A}\leq0
  • it is negative definite if det⁡A<0\det \mathrm{A}<0

An important consequence of these determinant-based definitions is that they show that all positive and negative definite matrices are invertible, since their determinant is never zero.