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In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. [31] The table at the right shows two possibilities for 2-by-2 matrices.
A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero ...
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" is only occasionally used due to the possible confusion with positive-definite matrices, which are different.
Totally positive matrix: A matrix with determinants of all its square submatrices positive. Triangular matrix: A matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular). Tridiagonal matrix
The matrix exponential of a real symmetric matrix is positive definite. Let be an n×n real symmetric matrix and a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,