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If two matrices of order can be multiplied in time (), where () for some >, then there is an algorithm computing the determinant in time (()). [53] This means, for example, that an O ( n 2.376 ) {\displaystyle \operatorname {O} (n^{2.376})} algorithm for computing the determinant exists based on the Coppersmith–Winograd algorithm .
For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation.
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 ...
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k) th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.
In one variable, the Hessian contains exactly one second derivative; if it is positive, then is a local minimum, and if it is negative, then is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the ...
Jacobian conjecture: Let k have characteristic 0. If J F is a non-zero constant, then F has an inverse function G: k N → k N which is regular, meaning its components are polynomials. According to van den Essen, [2] the problem was first conjectured by Keller in 1939 for the limited case of two variables and integer coefficients.
In particular, if has a positive determinant, then and can be chosen to be both rotations with reflections, or both rotations without reflections. [citation needed] If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.
Thus the determinant of a real skew-symmetric matrix is always non-negative. However this last fact can be proved in an elementary way as follows: 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 ...