MATLAB TUTORIAL, part 2.1: Positive Matrices

Positive Matrices
A square real matrix A is called positive-definite if all its eigenvalues are positive. In other words, if
\[ {\bf x}^{\mathrm T} \,{\bf A} \, {\bf x} >0 \]
for all nonzero real vectors x, and \( {\bf x}^{\ast} = {\bf x}^{\mathrm T} \) denotes the transposed vector. Correspondingly, a square matrix is called positive-semidefinite if all its eigenvalues are nonnegative:
\[ {\bf x}^{\ast} \,{\bf A} \, {\bf x} \ge 0 . \]
A square matrix A is called positive if all its entries are positive numbers. In particular, all Markov matrices are positive.

The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular (invertible). A positive definite matrix has at least one matrix square root. Furthermore, exactly one of its matrix square roots is itself positive definite. A real symmetric matrix A is positive definite iff there exists a real nonsingular matrix M such that

\[ {\bf A} = {\bf M}\, {\bf M}^{\mathrm T} . \]
Powers of square matrices will be denoted by exponents in the same manner as for scalars. For example, the square of a matrix A will be called \( {\bf A}^2 = {\bf A}\, {\bf A} , \) and it is equal to A times A. In a similar way, higher powers are defines as
\[ {\bf A}^n = {\bf A}\, {\bf A} \cdots {\bf A} = {\bf A}\, {\bf A}^{n-1} , \]
where the matrix A appears n times in the product on the right-hand side. The zero power of a square matrix is defined as the identity matrix, except when the matrix is null:
\[ {\bf A}^0 = {\bf I} , \]
where I is the identity matrix.