Assertion: matrix's determinant | /r/HasProperty | positive
Facets:
value: always
type: DEGREE
Source sentences:
text: Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
text: The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
text: The determinant can be a negative number.
text: the eigenvalues of a real skew-symmetric matrix are purely imaginary (see below) and to every eigenvalue there corresponds the conjugate eigenvalue with the same multiplicity; therefore, as the determinant is the product of the eigenvalues, each one repeated according to its multiplicity, it follows at once that the determinant, if it is not 0, is a positive real number.
text: The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
text: More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.
text: Its absolute value equals the area (in R 2 {\displaystyle \mathbb {R} ^{2}} ) or volume (in R 3 {\displaystyle \mathbb {R} ^{3}} ) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.
text: The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.
text: That is, the determinant of each submatrix of A obtained by deleting a set, possibly empty, of corresponding rows and columns of A is positive.
text: That is, the determinant of each submatrix of obtained by deleting a set, possibly empty, of corresponding rows and columns of is positive.
text: Thus the determinant of a real skew-symmetric matrix is always non-negative.