# Matrices¶

binomial

A binomial matrix that arose from the example in [bmsz01]. The matrix is a multiple of involutory matrix.

 [bmsz01] G. Boyd, C.A. Micchelli, G. Strang and D.X. Zhou, Binomial matrices, Adv. in Comput. Math., 14 (2001), pp 379-391.
cauchy

The Cauchy matrix is an m-by-n matrix with $$(i,j)$$ element

$\frac{1}{x_i - y_i}, \quad x_i - y_i \ne 0,$

where $$x_i$$ and $$y_i$$ are elements of vectors $$x$$ and $$y$$.

chebspec

Chebyshev spectral differentiation matrix. If k = 0,the generated matrix is nilpotent and a vector with all one entries is a null vector. If k = 1, the generated matrix is nonsingular and well-conditioned. Its eigenvalues have negative real parts.

chow

The Chow matrix is a singular Toeplitz lower Hessenberg matrix. The eigenvalues are known explicitly [chow69].

 [chow69] T.S. Chow, A class of Hessenberg matrices with known eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
circul

A circulant matrix has the property that each row is obtained by cyclically permuting the entries of the previous row one step forward.

clement

The Clement matrix [clem59] is a Tridiagonal matrix with zero diagonal entries. If k = 1, the matrix is symmetric.

 [clem59] P.A. Clement, A class of triple-diagonal matrices for test purposes, SIAM Review, 1 (1959), pp. 50-52.
companion

The companion matrix to a monic polynomial

$a(x) = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n$

is the n-by-n matrix with ones on the subdiagonal and the last column given by the coefficients of a(x).

dingdong

The Dingdong matrix is symmetric Hankel matrix invented by Dr. F. N. Ris of IBM, Thomas J Watson Research Centre. The eigenvalues cluster around $$\pi/2$$ and $$-\pi/2$$ [nash90].

 [nash90] J.C. Nash, Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, second edition, Adam Hilger, Bristol, 1990 (Appendix 1).
fiedler

The Fiedler matrix is symmetric matrix with a dominant positive eigenvalue and all the other eigenvalues are negative. For explicit formulas for the inverse and determinant, see [todd77].

 [todd77] J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, Birkhauser, Basel, and Academic Press, New York, 1977, pp. 159.
forsythe

The Forsythe matrix is a n-by-n perturbed Jordan block.

frank

The Frank matrix is an upper Hessenberg matrix with determinant 1. The eigenvalues are real, positive and very ill conditioned [vara86].

 [vara86] J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 835-839.
golub

Golub matrix is the product of two random unit lower and upper triangular matrices respectively. LU factorization without pivoting fails to reveal that such matrices are badly conditioned [vistre98].

 [vistre98] D. Viswanath and N. Trefethen. Condition Numbers of Random Triangular Matrices, SIAM J. Matrix Anal. Appl. 19, 564-581, 1998.
grcar

The Grcar matrix is a Toeplitz matrix with sensitive eigenvalues. The image below is a 200-by-200 Grcar matrix used in [nrt92].

 [nrt92] N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES algorithm for nonsymmetric linear system, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 796-825.

The Hadamard matrix is a square matrix whose entries are 1 or -1. It was named after Jacques Hadamard. The rows of a Hadamard matrix are orthogonal.

hankel

Hankel matrix is a a matrix that is symmetric and constant across the anti-diagonals. For example:

julia> matrixdepot("hankel", [1,2,3,4], [7,8,9,10])
4x4 Array{Float64,2}:
1.0  2.0  3.0   4.0
2.0  3.0  4.0   8.0
3.0  4.0  8.0   9.0
4.0  8.0  9.0  10.0

hilb

The Hilbert matrix is a very ill conditioned matrix. But it is symmetric positive definite and totally positive so it is not a good test matrix for Gaussian elimination [high02] (Sec. 28.1).

 [high02] (1, 2, 3, 4) Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, SIAM, PA, USA. 2002.
invhilb

Inverse of the Hilbert Matrix.

invol

An involutory matrix, i.e., a matrix that is its own inverse. See [hoca63].

 [hoca63] A.S. Householder and J.A. Carpenter, The singular values of involutory and idempotent matrices, Numer. Math. 5 (1963), pp. 234-237.
kahan

The Kahan matrix is a upper trapezoidal matrix, i.e., the $$(i,j)$$ element is equal to 0 if $$i > j$$. The useful range of theta is $$0 < theta < \pi$$. The diagonal is perturbed by pert*eps()*diagm([n:-1:1]).

kms

Kac-Murdock-Szego Toeplitz matrix [tren89].

 [tren89] W.F. Trench, Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., 10 (1989), pp. 135-146
lehmer

The Lehmer matrix is a symmetric positive definite matrix. It is totally nonnegative. The inverse is tridiagonal and explicitly known [neto58].

 [neto58] M. Newman and J. Todd, The evaluation of matrix inversion programs, J. Soc. Indust. Appl. Math, 6 (1958), pp. 466-476.
lotkin

The Lotkin matrix is the Hilbert matrix with its first row altered to all ones. It is unsymmetric, ill-conditioned and has many negative eigenvalues of small magnitude [lotk55].

 [lotk55] Lotkin, A set of test matrices, MTAC, 9, (1955), pp. 153-161.
magic

The magic matrix is a matrix with integer entries such that the row elements, column elements, diagonal elements and anti-diagonal elements all add up to the same number.

minij

A matrix with $$(i,j)$$ entry min(i,j). It is a symmetric positive definite matrix. The eigenvalues and eigenvectors are known explicitly. Its inverse is tridiagonal.

moler

The Moler matrix is a symmetric positive definite matrix. It has one small eigenvalue.

neumann

A singular matrix from the discrete Neumann problem. This matrix is sparse and the null space is formed by a vector of ones [plem76].

 [plem76] R.J. Plemmons, Regular splittings and the discrete Neumann problem, Numer. Math., 25 (1976), pp. 153-161.
oscillate

A matrix $$A$$ is called oscillating if $$A$$ is totally nonnegative and if there exists an integer q > 0 such that A^q is totally positive. An $$n \times n$$ oscillating matrix $$A$$ satisfies:

1. $$A$$ has $$n$$ distinct and positive eigenvalues $$\lambda_1 > \lambda_2 > \cdots > \lambda_n > 0$$.
2. The $$i$$ th eigenvector, corresponding to $$\lambda_i$$ in the above ordering, has exactly $$i -1$$ sign changes.

This function generates a symmetric oscillating matrix, which is useful for testing numerical regularization methods [hansen95]. For example:

julia> A = matrixdepot("oscillate", 3)
3x3 Array{Float64,2}:
0.98694    0.112794   0.0128399
0.112794   0.0130088  0.0014935
0.0128399  0.0014935  0.00017282

julia> eig(A)
([1.4901161192617526e-8,0.00012207031249997533,0.9999999999999983],
3x3 Array{Float64,2}:
0.0119607   0.113658  -0.993448
-0.215799   -0.969813  -0.113552
0.976365   -0.215743  -0.0129276)

 [hansen95] Per Christian Hansen, Test matrices for regularization methods. SIAM J. SCI. COMPUT Vol 16, No2, pp 506-512 (1995)
parter

The Parter matrix is a Toeplitz and Cauchy matrix with singular values near $$\pi$$ [part86].

 [part86] S. V. Parter, On the distribution of the singular values of Toeplitz matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130.
pascal

The Pascal matrix’s anti-diagonals form the Pascal’s triangle:

julia> matrixdepot("pascal", 6)
6x6 Array{Int64,2}:
1  1   1   1    1    1
1  2   3   4    5    6
1  3   6  10   15   21
1  4  10  20   35   56
1  5  15  35   70  126
1  6  21  56  126  252


See [high02] (28.4).

pei

The Pei matrix is a symmetric matrix with known inverse [pei62].

 [pei62] M.L. Pei, A test matrix for inversion procedures, Comm. ACM, 5 (1962), pp. 508.
poisson

A block tridiagonal matrix from Poisson’s equation. This matrix is sparse, symmetric positive definite and has known eigenvalues.

prolate

A prolate matrix is a symmetric ill-conditioned Toeplitz matrix

$\begin{split}A = \begin{bmatrix} a_0 & a_1 & \cdots \\ a_1 & a_0 & \cdots \\ \vdots & \vdots & \ddots \\ \end{bmatrix}\end{split}$

such that $$a_0= 2w$$ and $$a_k = (\sin 2 \pi wk)/\pi k$$ for $$k=1,2, \ldots$$ and $$0<w<1/2$$ [varah93].

 [varah93] J.M. Varah. The Prolate Matrix. Linear Algebra and Appl. 187:267–278, 1993.
randcorr

A random correlation matrix is a symmetric positive semidefinite matrix with 1s on the diagonal.

rando

A random matrix with entries -1, 0 or 1.

randsvd

Random matrix with pre-assigned singular values. See [high02] (Sec. 28.3).

rohess

A random orthogonal upper Hessenberg matrix. The matrix is constructed via a product of Givens rotations.

rosser

The Rosser matrix’s eigenvalues are very close together so it is a challenging matrix for many eigenvalue algorithms. matrixdepot("rosser", 8, 2, 1) generates the test matrix used in the paper [rlhk51]. matrixdepot("rosser") are more general test matrices with similar property.

 [rlhk51] Rosser, Lanczos, Hestenes and Karush, J. Res. Natl. Bur. Stand. Vol. 47 (1951), pp. 291-297. Archive
sampling

Matrices with application in sampling theory. A n-by-n nonsymmetric matrix with eigenvalues $$0, 1, 2, \ldots, n-1$$ [botr07].

 [botr07] L. Bondesson and I. Traat, A Nonsymmetric Matrix with Integer Eigenvalues, Linear and Multilinear Algebra, 55(3)(2007), pp. 239-247.
toeplitz

Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant. For example:

julia> matrixdepot("toeplitz", [1,2,3,4], [1,4,5,6])
4x4 Array{Int64,2}:
1  4  5  6
2  1  4  5
3  2  1  4
4  3  2  1

julia> matrixdepot("toeplitz", [1,2,3,4])
4x4 Array{Int64,2}:
1  2  3  4
2  1  2  3
3  2  1  2
4  3  2  1

tridiag

A group of tridiagonal matrices. matrixdepot("tridiagonal", n) generate a tridiagonal matrix with 1 on the diagonal and -2 on the upper- lower- diagonal, which is a symmetric positive definite M-matrix. This matrix is also known as Strang’s matrix, named after Gilbert Strang.

triw

Upper triangular matrices discussed by Wilkinson and others [gowi76].

 [gowi76] G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the computation of the Jordan canonical form, SIAM Review, 18(4), (1976), pp. 578-619.
vand

The Vandermonde matrix is defined in terms of scalars $$\alpha_0, \alpha_1, \ldots, \alpha_n$$ by

$\begin{split}V(\alpha_0, \ldots, \alpha_n) = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ \alpha_0 & \alpha_1 & \cdots & \alpha_n \\ \vdots & \vdots & & \vdots \\ \alpha_0^n & \alpha_1^n & \cdots & \alpha_n^n \\ \end{bmatrix}.\end{split}$

The inverse and determinant are known explicitly [high02].

wathen

The Wathen matrix is a sparse, symmetric positive, random matrix arising from the finite element method [wath87]. It is the consistent mass matrix for a regular nx-by-ny grid of 8-node elements.

 [wath87] A.J. Wathen, Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J. Numer. Anal., 7 (1987), pp. 449-457.
wilkinson

The Wilkinson matrix is a symmetric tridiagonal matrix with pairs of nearly equal eigenvalues. The most frequently used case is matrixdepot("wilkinson", 21).

Note

The images are generated using Winston.jl ‘s imagesc function.