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An n×n square matrix M with M_(ii) = 1 (1) M_(ij) = M_(ji)>1 (2) for all i,j=1, ..., n.

A cubic equation is an equation involving a cubic polynomial, i.e., one of the form a_3x^3+a_2x^2+a_1x+a_0=0. Since a_3!=0 (or else the polynomial would be quadratic and not ...

Given a square matrix M, the following are equivalent: 1. |M|!=0. 2. The columns of M are linearly independent. 3. The rows of M are linearly independent. 4. Range(M) = R^n. ...

A diagonal of a square matrix which is traversed in the "southeast" direction. "The" diagonal (or "main diagonal," or "principal diagonal," or "leading diagonal") of an n×n ...

The direction from an object A to another object B can be specified as a vector v=AB^-> with tail at A and head at B. However, since this vector has length equal to the ...

A generalization by Kronecker of Kummer's theory of prime ideal factors. A divisor on a full subcategory C of mod(A) is an additive mapping chi on C with values in a ...

If A is an n×n square matrix and lambda is an eigenvalue of A, then the union of the zero vector 0 and the set of all eigenvectors corresponding to eigenvalues lambda is ...

For an atomic integral domain R (i.e., one in which every nonzero nonunit can be factored as a product of irreducible elements) with I(R) the set of irreducible elements, the ...

Two matrices A and B are equal to each other, written A=B, if they have the same dimensions m×n and the same elements a_(ij)=b_(ij) for i=1, ..., n and j=1, ..., m. ...

The field axioms are generally written in additive and multiplicative pairs. name addition multiplication associativity (a+b)+c=a+(b+c) (ab)c=a(bc) commutativity a+b=b+a ...