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Let M_r be an r-rowed minor of the nth order determinant |A| associated with an n×n matrix A=a_(ij) in which the rows i_1, i_2, ..., i_r are represented with columns k_1, ...

If all the eigenvalues of a real matrix A have real parts, then to an arbitrary negative definite quadratic form (x,Wx) with x=x(t) there corresponds a positive definite ...

If mu=(mu_1,mu_2,...,mu_n) is an arbitrary set of positive numbers, then all eigenvalues lambda of the n×n matrix a=a_(ij) lie on the disk |z|<=m_mu, where ...

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, ...

The algebraic unknotting number of a knot K in S^3 is defined as the algebraic unknotting number of the S-equivalence class of a Seifert matrix of K. The algebraic unknotting ...

The operator norm of a linear operator T:V->W is the largest value by which T stretches an element of V, ||T||=sup_(||v||=1)||T(v)||. (1) It is necessary for V and W to be ...

Let beta=detB=x^2-ty^2, (1) where B is the Brahmagupta matrix, then det[B(x_1,y_1) B(x_2,y_2)] = det[B(x_1,y_1)]det[B(x_2,y_2)] (2) = beta_1beta_2]. (3)

An analog of the determinant for number triangles defined as a signed sum indexed by set partitions of {1,...,n} into pairs of elements. The Pfaffian is the square root of ...

A quaternion with complex coefficients. The algebra of biquaternions is isomorphic to a full matrix ring over the complex number field (van der Waerden 1985).

Each Cartan matrix determines a unique semisimple complex Lie algebra via the Chevalley-Serre, sometimes called simply the "Serre relations." That is, if (A_(ij)) is a k×k ...