rank.condition: Positive Definiteness of a Matrix, Rank and Condition Number

In corpcor: Efficient Estimation of Covariance and (Partial) Correlation

Description

is.positive.definite tests whether all eigenvalues of a symmetric matrix are positive.

make.positive.definite computes the nearest positive definite of a real symmetric matrix, using the algorithm of NJ Higham (1988) <DOI:10.1016/0024-3795(88)90223-6>.

rank.condition estimates the rank and the condition of a matrix by computing its singular values D[i] (using svd). The rank of the matrix is the number of singular values D[i] > tol) and the condition is the ratio of the largest and the smallest singular value.

The definition tol= max(dim(m))*max(D)*.Machine$double.eps is exactly compatible with the conventions used in "Octave" or "Matlab".

Also note that it is not checked whether the input matrix m is real and symmetric.

Usage

is.positive.definite(m, tol, method=c("eigen", "chol"))
make.positive.definite(m, tol)
rank.condition(m, tol)

Arguments

m	a matrix (assumed to be real and symmetric)
tol	tolerance for singular values and for absolute eigenvalues - only those with values larger than tol are considered non-zero (default: tol = max(dim(m))*max(D)*.Machine$double.eps)
method	Determines the method to check for positive definiteness: eigenvalue computation (eigen, default) or Cholesky decomposition (chol).

Value

is.positive.definite returns a logical value (TRUE or FALSE).

rank.condition returns a list object with the following components:

rank	Rank of the matrix.
condition	Condition number.
tol	Tolerance.

make.positive.definite returns a symmetric positive definite matrix.

Author(s)

Korbinian Strimmer (https://strimmerlab.github.io).

See Also

svd, pseudoinverse.

Examples

# load corpcor library
library("corpcor")

# Hilbert matrix
hilbert = function(n) { i = 1:n; 1 / outer(i - 1, i, "+") }

# positive definite ?
m = hilbert(8)
is.positive.definite(m)

# numerically ill-conditioned
m = hilbert(15)
rank.condition(m)

# make positive definite
m2 = make.positive.definite(m)
is.positive.definite(m2)
rank.condition(m2)
m2 - m

Example output

[1] TRUE
$rank
[1] 12

$condition
[1] Inf

$tol
[1] 6.148174e-15

[1] TRUE
$rank
[1] 15

$condition
[1] 1.502707e+14

$tol
[1] 6.148174e-15

      [,1] [,2] [,3]          [,4]          [,5]          [,6]          [,7]
 [1,]    0    0    0  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [2,]    0    0    0  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [3,]    0    0    0  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [4,]    0    0    0  0.000000e+00  0.000000e+00  1.387779e-17 -1.387779e-17
 [5,]    0    0    0  0.000000e+00  1.387779e-17 -5.551115e-17  1.526557e-16
 [6,]    0    0    0  1.387779e-17 -5.551115e-17  2.498002e-16 -6.661338e-16
 [7,]    0    0    0 -1.387779e-17  1.526557e-16 -6.661338e-16  1.804112e-15
 [8,]    0    0    0  2.775558e-17 -2.220446e-16  1.026956e-15 -2.872702e-15
 [9,]    0    0    0 -1.387779e-17  1.526557e-16 -7.355228e-16  2.220446e-15
[10,]    0    0    0 -1.387779e-17  5.551115e-17 -1.942890e-16  1.804112e-16
[11,]    0    0    0  1.387779e-17 -1.387779e-16  6.106227e-16 -1.484923e-15
[12,]    0    0    0  0.000000e+00 -6.938894e-18 -1.387779e-17  1.873501e-16
[13,]    0    0    0 -2.081668e-17  1.387779e-16 -5.689893e-16  1.276756e-15
[14,]    0    0    0  1.387779e-17 -1.040834e-16  4.371503e-16 -1.033895e-15
[15,]    0    0    0 -6.938894e-18  2.775558e-17 -1.040834e-16  2.567391e-16
               [,8]          [,9]         [,10]         [,11]         [,12]
 [1,]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [2,]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [3,]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00
 [4,]  2.775558e-17 -1.387779e-17 -1.387779e-17  1.387779e-17  0.000000e+00
 [5,] -2.220446e-16  1.526557e-16  5.551115e-17 -1.387779e-16 -6.938894e-18
 [6,]  1.026956e-15 -7.355228e-16 -1.942890e-16  6.106227e-16 -1.387779e-17
 [7,] -2.872702e-15  2.220446e-15  1.804112e-16 -1.484923e-15  1.873501e-16
 [8,]  4.829470e-15 -4.329870e-15  8.396062e-16  1.734723e-15 -6.245005e-16
 [9,] -4.329870e-15  5.211109e-15 -3.386180e-15  4.371503e-16  8.118506e-16
[10,]  8.396062e-16 -3.386180e-15  5.592748e-15 -4.399259e-15  5.065393e-16
[11,]  1.734723e-15  4.371503e-16 -4.399259e-15  6.036838e-15 -3.448630e-15
[12,] -6.245005e-16  8.118506e-16  5.065393e-16 -3.448630e-15  5.488665e-15
[13,] -1.304512e-15 -3.538836e-16  1.873501e-15  1.873501e-16 -4.496403e-15
[14,]  1.221245e-15 -9.020562e-17 -1.387779e-15  6.661338e-16  1.942890e-15
[15,] -3.191891e-16  6.245005e-17  3.191891e-16 -2.220446e-16 -3.469447e-16
              [,13]         [,14]         [,15]
 [1,]  0.000000e+00  0.000000e+00  0.000000e+00
 [2,]  0.000000e+00  0.000000e+00  0.000000e+00
 [3,]  0.000000e+00  0.000000e+00  0.000000e+00
 [4,] -2.081668e-17  1.387779e-17 -6.938894e-18
 [5,]  1.387779e-16 -1.040834e-16  2.775558e-17
 [6,] -5.689893e-16  4.371503e-16 -1.040834e-16
 [7,]  1.276756e-15 -1.033895e-15  2.567391e-16
 [8,] -1.304512e-15  1.221245e-15 -3.191891e-16
 [9,] -3.538836e-16 -9.020562e-17  6.245005e-17
[10,]  1.873501e-15 -1.387779e-15  3.191891e-16
[11,]  1.873501e-16  6.661338e-16 -2.220446e-16
[12,] -4.496403e-15  1.942890e-15 -3.469447e-16
[13,]  5.676015e-15 -3.046174e-15  6.314393e-16
[14,] -3.046174e-15  1.755540e-15 -3.747003e-16
[15,]  6.314393e-16 -3.747003e-16  8.326673e-17

corpcor documentation built on Sept. 16, 2021, 5:12 p.m.