As seen in class, there are many different matrices with special properties that are named after people (Wikipedia list) such as the Vandermonde matrix. Another website has a relatively comprehensive list of special matrices and matrix properties. The ones described below are particularly well-known, have especially interesting properties, and can be handily generated by functions in Matlab.

Hilbert Matrices – hilb(n) - A square matrix with elements i,j = 1 / (i+j-1).
Interesting Properties: Symmetric. Famously ill-conditioned matrix - numerical computations performed with Hilbert matrices are difficult and generally result in large errors. Often used as test matrices for numerical analysis methods such as linear system solvers.

e.g. a 5×5 Hilbert matrix
1.0000 0.5000 0.3333 0.2500 0.2000
0.5000 0.3333 0.2500 0.2000 0.1667
0.3333 0.2500 0.2000 0.1667 0.1429
0.2500 0.2000 0.1667 0.1429 0.1250
0.2000 0.1667 0.1429 0.1250 0.1111

Rosser Matrix - rosser() - an 8×8 matrix with the following entries:

611 196 -192 407 -8 -52 -49 29
196 899 113 -192 -71 -43 -8 -44
-192 113 899 196 61 49 8 52
407 -192 196 611 8 44 59 -23
-8 -71 61 8 411 -599 208 208
-52 -43 49 44 -599 411 208 208
-49 -8 8 59 208 208 99 -911
29 -44 52 -23 208 208 -911 99

Interesting Properties: Classic test matrix used on matrix eigenvalue algorithms. The matrix has a characteristic polynomial that factors into x (x - 1020) (x^2 - 1020 x + 100)(x^2 - 1040500) (x - 1000) ^2, with eigenvalues 0, 1020, 10*10405^(1/2), -10*10405^(1/2), 510+100*26^(1/2), 510-100*26^(1/2), 1000, 1000.

Wilkinson Matrices – wilkinson(n) - J. H. Wilkinson’s test matrices used on matrix eigenvalue algorithms.
Interesting Properties: the matrices are symmetric, tridiagonal, and have pairs of eigenvalues that are very close to each other. The most often used test case is wilkinson(21), with the largest eigenvalue pairs 10.74619418290332 and 10.74619418290340.

e.g. the 5×5 Wilkinson Matrix:

2 1 0 0 0
1 1 1 0 0
0 1 0 1 0
0 0 1 1 1
0 0 0 1 2

Hadamard matrices – hadamard(n) - A square NxN matrix H with elements consisting of only 1 or -1 such that H*H’ = N*I, where I is the NxN identity matrix.
Interesting Properties: Each row in the matrix is orthogonal with every other row. det(H) = N^(N/2). Has applications in error detecting and correcting codes.

e.g. a 8×8 Hadamard matrix:

1 1 1 1 1 1 1 1
1 -1 1 -1 1 -1 1 -1
1 1 -1 -1 1 1 -1 -1
1 -1 -1 1 1 -1 -1 1
1 1 1 1 -1 -1 -1 -1
1 -1 1 -1 -1 1 -1 1
1 1 -1 -1 -1 -1 1 1
1 -1 -1 1 -1 1 1 -1

Hankel matrices – hankel(c,r) - A square NxN matrix H with elements Ai,j = Ai-1,j+1.
Interesting Properties: Entries are constant across skew-diagonals, so the matrix only depends upon the first column vector and last row vector. Symmetric. The Hankel transform of order zero is equivalent to a 2-D Fourier transform and can be performed easily for circularly symmetric functions.

e.g. 5×5 Hankel matrix

Toeplitz matrices – toeplitiz(c,r) - A square NxN matrix T with elements Ai,j = Ai-1,j-1. This is the Hankel matrix upside-down.
Interesting Properties: Entries are constant across diagonals, so the matrix only depends upon the first column vector and last row vector. Solving numerical problems involving Toeplitz matrices are easier since the matrix can be defined by 2n-1 instead of n^2 values. For example, two Toeplitz matrices can be added together in O(n), and multiplied in O(n^2).

e.g. 5×5 Toeplitz matrix

Vandermonde matrices – vander(v) - A matrix V with elements Vi,j = vi^(j-1)
Interesting Properties: As mentioned in lecture, in polynomial interpolation, the problem of solving for the coefficients c[1..n] of a polynomial given x[1..n] and y[1..n] can be represented in matrix form as solving Vc = y. Additionally, the Discrete Fourier Transform can be represented by a Vandermonde matrix.

Pascal matrices – pascal(n,t) - Symmetric, positive definite matrices with elements as the Pascal triangle with appropriate truncation, in three possible ways – as an upper-triangular matrix L, a lower-triangular matrix U, or a symmetric matrix S.
Interesting Properties: S = LU. The inverse of S, L, U all have entirely positive entries. The Cholesky factor matrix of S is its own inverse. The determinants of the Pascal matrices as well as the Cholesky factors of S are all 1.

e.g. a 5×5 Pascal matrix
L =

1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1

U =

1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1

S =

1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70

with Cholesky factor

1 0 0 0 0
1 -1 0 0 0
1 -2 1 0 0
1 -3 3 -1 0
1 -4 6 -4 1