Positive-definite Matrix - Examples

Examples

  • The identity matrix is positive definite, by any definition. Seen as a real matrix, it is symmetric, and, for any column vector z with real entries a and b, one has . Seen as complex matrix, for any column vector z with complex entries a and b one has . Either way, the result is positive as long as z is not the zero vector (that is, at least one of a and b is not zero).
  • The real symmetric matrix is positive definite since for any non-zero column vector z with entries a, b and c, we have
\begin{align} z^{\mathrm{T}}M z &= \begin{bmatrix} (2a-b)&(-a+2b-c)&(-b+2c) \end{bmatrix} \begin{bmatrix} a\\b\\c \end{bmatrix} \\ &= 2{a}^2 - 2ab + 2{b}^2 - 2bc + 2{c}^2 \\ &= {a}^2+(a - b)^{2} + (b - c)^{2}+{c}^2 \end{align}
This result is a sum of squares, and therefore non-negative; and is zero only if, that is, when z is zero.
  • The real symmetric matrix is not positive definite. If z is the vector, one has

The examples M and N above show that a matrix in which some elements are negative may still be positive-definite, and conversely a matrix whose entries are all positive may not be positive definite.

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