binomialcauchychebspecchowcirculclementcompaniondingdongfiedlerforsythefrankgolubgrcarhadamardhankelhilbinvhilbinvolkahankmslehmerlotkinmagicminijmolerneumannoscillateparterpascalpeipoissonprolaterandcorrrandorandsvdrohessrossersamplingtoeplitztridiagtriwvandwathenwilkinson
- companion
The companion matrix to a monic polynomial
a(x) = a0 + a1x + ⋯ + an − 1xn − 1 + xnis the n-by-n matrix with ones on the subdiagonal and the last column given by the coefficients of a(x).
- 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].
- 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- 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- prolate
A prolate matrix is a symmetric ill-conditioned Toeplitz matrix
- A = begin{bmatrix}
- a_0 & a_1 & cdots \
a_1 & a_0 & cdots \ vdots & vdots & ddots \ end{bmatrix}
such that a0 = 2w and ak = (sin 2πwk)/πk for k = 1, 2, … and 0 < w < 1/2 [varah93].
- 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 × n oscillating matrix A satisfies:
- A has n distinct and positive eigenvalues
- λ1 > λ2 > ⋯ > λn > 0.
- The i th eigenvector, corresponding to λi 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)
- 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.
- kms
Kac-Murdock-Szego Toeplitz matrix [tren89].
- rohess
A random orthogonal upper Hessenberg matrix. The matrix is constructed via a product of Givens rotations.
- randsvd
Random matrix with pre-assigned singular values. See [high02] (Sec. 28.3).
- rando
A random matrix with entries -1, 0 or 1.
- wilkinson
The Wilkinson matrix is a symmetric tridiagonal matrix with pairs of nearly equal eigenvalues. The most frequently used case is
matrixdepot("wilkinson", 21).
- 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].
- poisson
A block tridiagonal matrix from Poisson's equation. This matrix is sparse, symmetric positive definite and has known eigenvalues.
- randcorr
A random correlation matrix is a symmetric positive semidefinite matrix with 1s on the diagonal.
- chow
The Chow matrix is a singular Toeplitz lower Hessenberg matrix. The eigenvalues are known explicitly [chow69].
- parter
The Parter matrix is a Toeplitz and Cauchy matrix with singular values near π [part86].
- lehmer
The Lehmer matrix is a symmetric positive definite matrix. It is totally nonnegative. The inverse is tridiagonal and explicitly known [neto58].
- 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.
- binomial
A binomial matrix that arose from the example in [bmsz01]. The matrix is a multiple of involutory matrix.
- 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.
- clement
The Clement matrix [clem59] is a Tridiagonal matrix with zero diagonal entries. If
k = 1, the matrix is symmetric.
- 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].
- 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].
- chebspec
Chebyshev spectral differentiation matrix. If
k = 0,the generated matrix is nilpotent and a vector with all one entries is a null vector. Ifk = 1, the generated matrix is nonsingular and well-conditioned. Its eigenvalues have negative real parts.
- invol
An involutory matrix, i.e., a matrix that is its own inverse. See [hoca63].
- vand
The Vandermonde matrix is defined in terms of scalars α0, α1, …, αn by
$$\begin{aligned} V(\alpha_0, \ldots, \alpha_n) = \begin{bmatrix} 1 & 1 & \cdots & 1 \\\ \alpha_0 & \alpha_1 & \cdots & \alpha_n \\\ \vdots & \vdots & & \vdots \\\ \alpha_0^n & \alpha_1^n & \cdots & \alpha_n^n \\\ \end{bmatrix}. \end{aligned}$$ The inverse and determinant are known explicitly [high02].
- pei
The Pei matrix is a symmetric matrix with known inversison [pei62].
- 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
thetais 0 < theta < π. The diagonal is perturbed bypert*eps()*diagm([n:-1:1]).
- 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).
- sampling
Matrices with application in sampling theory. A n-by-n nonsymmetric matrix with eigenvalues 0, 1, 2, …, n − 1 [botr07].
- moler
The Moler matrix is a symmetric positive definite matrix. It has one small eigenvalue.
- triw
Upper triangular matrices discussed by Wilkinson and others [gowi76].
- forsythe
The Forsythe matrix is a n-by-n perturbed Jordan block.
- 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 xi and yi are elements of vectors x and y.
- 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.
- hadamard
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.
- 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 π/2 and − π/2 [nash90].
- invhilb
Inverse of the Hilbert Matrix.
- grcar
The Grcar matrix is a Toeplitz matrix with sensitive eigenvalues. The image below is a 200-by-200 Grcar matrix used in [nrt92].
- frank
The Frank matrix is an upper Hessenberg matrix with determinant 1. The eigenvalues are real, positive and very ill conditioned [vara86].
- circul
A circulant matrix has the property that each row is obtained by cyclically permuting the entries of the previous row one step forward.
- 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.
- 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).
Note
The images are generated using Winston.jl 's imagesc function.