Mathematical Notation

Descriptions of LAPACK routines use the following notation:


For an M-by-N matrix A, denotes the conjugate transposed N-by-M matrix with elements:

For a real-valued matrix, AH = AT.


The dot product of two vectors, defined as:

Ax = b

A system of linear equations with an n-by-n matrix A = {aij}, a right-hand side vector b = {bi}, and an unknown vector x = {xi}.

AX = B

A set of systems with a common matrix A and multiple right-hand sides. The columns of B are individual right-hand sides, and the columns of X are the corresponding solutions.


the vector with elements |xi| (absolute values of xi).


the matrix with elements |aij| (absolute values of aij).

||x|| = maxi|xi|

The infinity-norm of the vector x.

||A|| = maxiΣj|aij|

The infinity-norm of the matrix A.

||A||1 = maxjΣi|aij|

The one-norm of the matrix A. ||A||1 = ||AT|| = ||AH||


The 2-norm of the vector x: ||x||2 = (Σi|xi|2)1/2 = ||x||E (see the definition for Euclidean norm in this topic).


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


The Euclidean norm of the matrix A: ||A||E2 = ΣiΣj|aij|2.

κ(A) = ||A||·||A-1||

The condition number of the matrix A.


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


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

For more complete information about compiler optimizations, see our Optimization Notice.