Accession

Date on which the matrix has been made publically available for the first time.

Block Triangular Form

A rectangular matrix has a block triangular form (BTF) if there exists a row and a column permutation such that the permuted matrix has the form on the right side of Figure (coarse decomposition). Au is underdetermined, As is square, Ao is overdetermined, and X’s denote possibly nonzero matrices of appropriate dimensions.
One or two of the three submatrices may be absent in the coarse decomposition. Au,Ao may be decomposable into block diagonal form. The square submatrix As may have a block upper triangular form. The further decomposition of the submatrices Au,As and Ao is called fine decomposition (on the right of Figure). Note that if the overdetermined block Ao appears in the coarse decomposition of a matrix, then the matrix is structurally rank deficient.
Algorithms for computing the BTF of a sparse matrix are based on a canonical decomposition of bipartite graphs discovered by Dulmage and Mendelsohn. We use the code of Pothen and Fan.

PIC

Block triangular form of a rectangular matrix.

Conditioning

1 norm
∥A∥1 = maxj∑ i∣aij∣
2 norm
∥A∥2 =
Infinity norm
∑ ∥A ∥∞ = maxi | aij | j

Frobenius norm
∘ ∑---∑---------- ∥A∥F = | aij |2 i j

Definiteness

positive definite
A Hermitian matrix A is positive definite if ∀x≠0 xHAx > 0. The diagonal elements of a positive definite matrix are positive. A positive definite matrix can be factored uniquely in the form A = RHR, where R is upper triangular with positive diagonal elements. The requirement that A be Hermitian reduces to symmetry for real matrices (AH = AT ).
positive semidefinite
A Hermitian matrix A is positive semidefinite if ∀x≠0 xHAx ≥ 0. The requirement that A be Hermitian reduces to symmetry for real matrices.
negative definite
A Hermitian matrix A is negative definite if ∀x≠0 xHAx < 0. The requirement that A be Hermitian reduces to symmetry for real matrices.
negative semidefinite
A Hermitian matrix A is negative semidefinite if ∀x≠0 xHAx ≤ 0. The requirement that A be Hermitian reduces to symmetry for real matrices.

Diagonal dominance

diagonally dominant by rows
A square matrix A is diagonally dominant by rows if
∑ |aii| ≥ |aij|,∀i j⁄=i
strictly diagonally dominant by rows
A square matrix A is strictly diagonally dominant by rows if
∑ |aii| > |aij|,i = 1, ⋅⋅⋅n j⁄=i
A strictly diagonally dominant matrix is nonsingular. Gaussian elimination without pivoting on a matrix strictly diagonal dominant by rows is stable.
diagonally dominant by columns
A square matrix A is diagonally dominant by columns if AT is diagonally dominant by rows.
strictly diagonally dominant by columns
A square matrix A is strictly diagonally dominant by columns if AT is strictly diagonally dominant by rows. A strictly diagonally dominant matrix is nonsingular. Gaussian elimination without pivoting on a matrix strictly diagonal dominant by columns is stable.

Entries

Nb Entries in File
is the number of values stored in a matrix file. Note that this may include explicit zeros, and that for matrices with symmetric pattern (i.e. with the attribut symmetry equal to symmetric, Hermitian, skew symmetric) the entries in the lower triangular part (strict lower triangular for skew symmetric) of the matrix are stored in the file. For what concern duplicates and out of range entries see duplicates and out of range .
Entries
is the total number of values in the matrix including explicit zeros. For matrices with symmetric pattern (i.e. with the attribut symmetry equal to symmetric, Hermitian, skew symmetric) this is different from Nb Entries in File. Attribut only for public matrices.
Non zeros
Total number of numerical values in the matrix different from zero. Attribut only for public matrices.
Explicit zeros
Total number of numerical values in the matrix equal to zero. Attribut only for public matrices.
Duplicates
Entries that have same indices i,j. Duplicates entries are allowed in matrices of private groups, but not for matrices in the Public group. If present in matrices uploaded in the public group, duplicates will be summed together.
Out of range
Entries whose indices are ≤ 0 or greater than n for colum indices and greater than m, for row indices. Note that most of the software factorizing sparse matrices don’t manage out of range entries.

Format

Harwell Boeing
See Users’ Guide for the Harwell-Boeing Sparse Matrix Collection for the complete description of this format.
Rutherford Boeing
See The rutherford-Boeing Sparse Matrix Collection for the complete description of this format.
Matrix Market
See The Matrix Market Exchange Formats : Initial Design for the complete description of this format.
TLSE

Rank

Numerical
The column rank (row rank) of a matrix A is the maximal number of linearly independent columns (rows) of A. The column rank and the row rank are always equal, therefore, they are simply called the rank of A. The rank of an mxn matrix is at most min(m,n). A matrix that has a rank as large as possible is said to have full rank ; otherwise, the matrix is rank deficient.
Structural
Computed the block triangular form of A, the structural rank is given by
N b rows of Au + N b rows of As + N b columns of Ao

Reducibility

A sparse matrix is reducible if there exist row and colum permutations such that the permuted matrix has a block triangular form , with irreducible diagonal blocks.

Source

The person at the origin of the matrix.

Storage Mode

assembled format
matrices of the form A = [aij], where aij is a single entry of the matrix are called assembled matrices.
Nb Columns For an assembled mxn matrix this corresponds to the dimension n of the matrix.
Nb Rows For an assembled mxn matrix this corresponds to the dimension m of the matrix.
elemental format
matrices of the form A = ∑ Aij, where Aij is a small dense matrix, are called unassembled matrices, and the Aij matrices elemental matrices. Very often the term elemental matrix is used for the whole matrix A and element for the single Aij matrix. Elemental matrices arise in finite element applications.
Nb Elements corresponds to the total number of elemental matrices present in the file.
Max Var corresponds to the maximum row index present in the file.
Nb Variable Indices corresponds to the total number of variables indices present in the file.

Submitter

The person who uploaded the file to the Grid-TLSE site.

Symmetry

symmetric
A square matrix A is symmetric if A = AT where AT is the transpose of A : aij = aji. Symmetric matrices have the same elements above the diagonal as below. In all formats used on this site, only the entries of the lower triangle of a symmetric matrix are stored in the file.
Hermitian
A square complex matrix A is Hermitian if A = AH where AH is the complex coniugate of the transpose of A ; that is aij = Żaji. It follows that the diagonal elements of a Hermitian matrix are real. In all formats used on this site, only the entries of the lower triangle of a Hermitian matrix are stored in the file.
skew-symmetric
A square matrix A is skew-symmetric if A = -AT where AT is the transpose of A : aij = -aji. The diagonal elements of a skew-symmetric matrix are zero. In all formats used on this site, only the entries of the strict lower triangle of a skew-symmetric matrix are stored in the file.
unsymmetric
A square matrix A is unsymmetric if no relation can be establish between the elements above the diagonal with those below. Note that an unsymmetric matrix can have a symmetric pattern (i.e. for each (i,j) there exists an element in position (j,i)) but the values of the elements are different aij≠aji.
rectangular
A mxn matrix A is rectangular if m≠n.

Rate of Symmetry

It is computed only for unsymmetric matrices. For matrices symmetric in pattern (symmetric, Hermitian, skew symmetric) the rate of symmetry is 100%.
Structural symmetry
Given an unsymmetric matrix A = [aij], let
match = {(i,j ),(j,i) | a and a ∈ A } i,j j,i
Note that if a diagonal entry is present in A (ai,i ∈ A), it is counted only once. We define structural symmetry of a matrix
∥match ∥ ∕(Entries in A)
in pourcentage. Note that a diagonal matrix has a structural symmetry = 100%
Numerical symmetry (with zeros)
The structural symmetry is computed and a test of equality is made on the numerical values.
Numerical symmetry (without zeros)
The structural symmetry is computed and a test of equality is made on the numerical values different from zero.

Type Values

complex
the numerical values are complex numbers.
pattern
no numerical values supplied
real
the numerical values are real numbers.

Values

Max value
For a matrix A = [aij], max value = max i,j∣aij∣, where ∣aij∣, means the absolute of aij, if aij ∈ ℝ, otherwise the modulus if aij ∈ ℂ.
Min value
For a matrix A = [aij], min value = min i,j∣aij∣ where ∣aij∣, means the absolute of aij, if aij ∈ ℝ, otherwise the modulus if aij ∈ ℂ.