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A real matrix is a matrix whose elements consist entirely of real numbers. The set of m×n real matrices is sometimes denoted R^(m×n) (Zwillinger 1995, p. 116).

For a given monic quartic equation f(x)=x^4+a_3x^3+a_2x^2+a_1x+a_0, (1) the resolvent cubic is the monic cubic polynomial g(x)=x^3+b_2x^2+b_1x+b_0, (2) where the coefficients ...

A right eigenvector is defined as a column vector X_R satisfying AX_R=lambda_RX_R. In many common applications, only right eigenvectors (and not left eigenvectors) need be ...

The direct product of the rings R_gamma, for gamma some index set I, is the set product_(gamma in I)R_gamma={f:I-> union _(gamma in I)R_gamma|f(gamma) in R_gamma all gamma in ...

If the matrices A, X, B, and C satisfy AX-XB=C, then [I X; 0 I][A C; 0 B][I -X; 0 I]=[A 0; 0 B], where I is the identity matrix.

A scalar is a one-component quantity that is invariant under rotations of the coordinate system.

A map f:R^n|->R which assigns each x a scalar function f(x).

A function f(x_1,...,x_n) of one or more variables whose range is one-dimensional, as compared to a vector function, whose range is three-dimensional (or, in general, ...

A diagonal matrix whose diagonal elements all contain the same scalar lambda. A scalar matrix is therefore equivalent to lambdaI, where I is the identity matrix.

Let A=a_(ij) be an n×n matrix with complex (or real) entries and eigenvalues lambda_1, lambda_2, ..., lambda_n, then sum_(i=1)^n|lambda_i|^2<=sum_(i,j=1)^n|a_(ij)|^2 (1) ...