Mathematical Notation for LAPACK Routines

Developer Reference for Intel® oneAPI Math Kernel Library for C

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Date 4/28/2026

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Mathematical Notation for LAPACK Routines

Descriptions of LAPACK routines use the following notation:

`\(A^{H}\)`	For an `\(m\)`-by-`\(n\)` matrix `\(A\)`, denotes the conjugate transposed `\(n\)`-by-`\(m\)` matrix with elements: \[a_{ij}^H = \overline{a_{ji}}\] For a real-valued matrix, `\(A^H = A^T\)`.
`\(x \cdot y\)`	The dot product of two vectors, defined as: \[x \cdot y = \sum_i x_i \overline{y_i}\]
`\(A \cdot x = b\)`	A system of linear equations with an `\(n\)`-by-`\(n\)` matrix `\(A = \{a_{ij}\}\)`, a right-hand side vector `\(b = \{b_i\}\)`, and an unknown vector `\(x = \{x_i\}\)`.
`\(A \cdot X = B\)`	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.
`\(\lvert x \rvert\)`	The vector with elements `\(\lvert x_i \rvert\)` (absolute values of `\(x_i\)`).
`\(\lvert A \rvert\)`	The matrix with elements `\(\lvert a_{ij} \rvert\)` (absolute values of `\(a_{ij}\)`).
`\({\lVert x \rVert}_{\infty} = \max_i \lvert x_i \rvert\)`	The infinity-norm of the vector `\(x\)`.
`\({\lVert A \rVert}_{\infty} = \max_i \sum_j \lvert a_{ij} \rvert\)`	The infinity-norm of the matrix `\(A\)`.
`\({\lVert A \rVert}_1 = \max_j \sum_i \lvert a_{ij} \rvert\)`	The one-norm of the matrix `\(A\)`. `\({\lVert A \rVert}_1 = {\lVert A^T \rVert}_{\infty} = {\lVert A^H \rVert}_{\infty}\)`
`\({\lVert x \rVert}_2\)`	The 2-norm of the vector `\(x\)` : `\({\lVert x \rVert}_2 = (\sum_i {\lvert x_i \rvert}^2)^{1/2} = {\lVert x \rVert}_E\)` (see the definition for Euclidean norm in this topic).
`\({\lVert A \rVert}_2\)`	The 2-norm (or spectral norm) of the matrix `\(A\)`. `\({\lVert A \rVert}_2 = max_i \sigma_i\)`; `\({\lVert A \rVert}_2^2 = max_{{\lVert x \rVert}_2 = 1}(Ax \cdot Ax)\)`
`\({\lVert A \rVert}_E\)`	The Euclidean norm of the matrix `\(A\)` : `\({\lVert A \rVert}_E^2 = \sum_i \sum_j {\lvert a_{ij} \rvert}^2\)`.
`\(\kappa(A) = {\lVert A \rVert} \cdot {\lVert A^{-1} \rVert}\)`	The condition number of the matrix `\(A\)`.
`\(\lambda_i\)`	Eigenvalues of the matrix `\(A\)` (for the definition of eigenvalues, see Eigenvalue Problems).
`\(\sigma_i\)`	Singular values of the matrix `\(A\)`. They are equal to square roots of the eigenvalues of `\(A^H A\)`. (For more information, see Singular Value Decomposition).