Mathematical Notation

In addition to the mathematical notation used in description of BLAS and LAPACK Linear Equations routines, descriptions of the routines to solve Least Squares and Eigenvalue plroblems use the following notation:

λ_i

Eigenvalues of the matrix A (for the definition of eigenvalues, see Eigenvalue Problems).

σ_i

Singular values of the matrix A. They are equal to square roots of the eigenvalues of A^HA. (For more information, see Singular Value Decomposition).

||x||₂

The 2-norm of the vector x: ||x||₂ = (Σ_i|x_i|²)^1/2 = ||x||_E .

||A||₂

The 2-norm (or spectral norm) of the matrix A.

||A||₂ = max_iσ_i, ||A||₂²= max_|x|=1(Ax·Ax).

||A||_E

The Euclidean norm of the matrix A: ||A||_E² = Σ_iΣ_j|a_ij|² (for vectors, the Euclidean norm and the 2-norm are equal: ||x||_E = ||x||₂).

q(x, y)

The acute angle between vectors x and y:

cos q(x, y) = |x·y| / (||x||₂||y||₂).