Is a matrix symmetric or positive-definite?

Description

Determines if a matrix is square, symmetric, positive-definite, or positive semi-definite.

Usage

is.square.matrix(A)

is.symmetric.matrix(A, tol = .Machine$double.eps^0.5)

is.positive.semi.definite(A, tol = .Machine$double.eps^0.5)

is.positive.definite(A, tol = .Machine$double.eps^0.5)

Arguments

Details

A tolerance is added to indicate if a matrix A is approximately symmetric. If A is not symmetric, a message and first few rows of the matrix is printed. If A has any missing values, NA is returned.

• is.symmetric.matrix returns TRUE if A is a numeric, square and symmetric matrix; otherwise, returns FALSE. A matrix is symmetric if the absolute difference between A and its transpose is less than tol.

• is.positive.semi.definite returns TRUE if a real, square, and symmetric matrix A is positive semi-definite. A matrix is positive semi-definite if its smallest eigenvalue is greater than or equal to zero.

• is.positive.definite returns TRUE if a real, square, and symmetric matrix A is positive-definite. A matrix is positive-definite if its smallest eigenvalue is greater than zero.

Note

Functions are adapted from Frederick Novomestky's matrixcalc package in order to implement the rmatnorm function. The following changes are made:

• I changed argument x to A to reflect usual matrix notation.

• For is.symmetric, I added a tolerance so that A is symmetric even provided small differences between A and its transpose. This is useful for rmatnorm function, which was used repeatedly to generate matrixNormal random variates in a Markov chain.

• For is.positive.semi.definite and is.positive.definite, I also saved time by avoiding a for-loop and instead calculating the minimum of eigenvalues.

Examples

## Example 0: Not square matrix
B <- matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, byrow = TRUE)
B
is.square.matrix(B)

## Example 1: Not a matrix. should get an error.
df <- as.data.frame(matrix(c(1, 2, 3, 4, 5, 6), nrow = 2, byrow = TRUE))
df
## Not run: 
is.square.matrix(df)

## End(Not run)

## Example 2: Not symmetric & compare against matrixcalc
F <- matrix(c(1, 2, 3, 4), nrow = 2, byrow = TRUE)
F
is.square.matrix(F)
is.symmetric.matrix(F) # should be FALSE
if (!requireNamespace("matrixcalc", quietly = TRUE)) {
  matrixcalc::is.symmetric.matrix(F)
} else {
  message("you need to install the package matrixcalc to compare this example")
}

## Example 3: Symmetric but negative-definite. The functions are same.
# eigenvalues are  3 -1
G <- matrix(c(1, 2, 2, 1), nrow = 2, byrow = TRUE)
G
is.symmetric.matrix(G)
if (!requireNamespace("matrixcalc", quietly = TRUE)) {
  matrixcalc::is.symmetric.matrix(G)
} else {
  message("you need to install the package matrixcalc to compare this example.")
}
isSymmetric.matrix(G)
is.positive.definite(G) # FALSE
is.positive.semi.definite(G) # FALSE

## Example 3b: A missing value in G
G[1, 1] <- NA
is.symmetric.matrix(G) # NA
is.positive.definite(G) # NA

## Example 4: positive definite matrix
# eigenvalues are 3.4142136 2.0000000 0.585786
Q <- matrix(c(2, -1, 0, -1, 2, -1, 0, -1, 2), nrow = 3, byrow = TRUE)
is.symmetric.matrix(Q)
is.positive.definite(Q)

## Example 5: identity matrix is always positive definite
I <- diag(1, 3)
is.square.matrix(I) # TRUE
is.symmetric.matrix(I) # TRUE
is.positive.definite(I) # TRUE

The Matrix Normal Distribution

Description

Computes the density (dmatnorm), calculates the cumulative distribution function (CDF, pmatnorm), and generates 1 random number (rmatnorm) from the matrix normal:

A \sim MatNorm_{n,p}(M, U, V)

.

Usage

dmatnorm(A, M, U, V, tol = .Machine$double.eps^0.5, log = TRUE)

pmatnorm(
  Lower = -Inf,
  Upper = Inf,
  M,
  U,
  V,
  tol = .Machine$double.eps^0.5,
  keepAttr = TRUE,
  algorithm = mvtnorm::GenzBretz(),
  ...
)

rmatnorm(s = 1, M, U, V, tol = .Machine$double.eps^0.5, method = "chol")

Arguments

Details

These functions rely heavily on this following property of matrix normal distribution. Let koch() refer to the Kronecker product of a matrix. For a n x p matrix A, if

A \sim MatNorm(M, U, V),

then

vec(A) \sim MVN_{np} (M, Sigma = koch(V,U) ).

Thus, the probability of Lower < A < Upper in the matrix normal can be found by using the CDF of vec(A), which is given by pmvnorm function in mvtnorm. See algorithms and pmvnorm for more information.

Also, we can simulate a random matrix A from a matrix normal by sampling vec(A) from rmvnorm function in mvtnorm. This matrix A takes the rownames from U and the colnames from V.

Calculating Matrix Normal Probabilities

From the mvtnorm package, three algorithms are available for evaluating normal probabilities:

• The default is the randomized Quasi-Monte-Carlo procedure by Genz (1992, 1993) and Genz and Bretz (2002) applicable to arbitrary covariance structures and dimensions up to 1000.

• For smaller dimensions (up to 20) and non-singular covariance matrices, the algorithm by Miwa et al. (2003) can be used as well.

• For two- and three-dimensional problems and semi-infinite integration region, TVPACK implements an interface to the methods described by Genz (2004).

The ... arguments define the hyper-parameters for GenzBertz algorithm:

maxpts

maximum number of function values as integer. The internal FORTRAN code always uses a minimum number depending on the dimension.Default 25000.

abseps

absolute error tolerance.

releps

relative error tolerance as double.

Note

Ideally, both scale matrices are positive-definite. If they do not appear to be symmetric, the tolerance should be increased. Since symmetry is checked, the 'checkSymmetry' arguments in 'mvtnorm::rmvnorm()' are set to FALSE.

References

Pocuca, N., Gallaugher, M.P., Clark, K.M., & McNicholas, P.D. (2019). Assessing and Visualizing Matrix Variate Normality. Methodology. <https://arxiv.org/abs/1910.02859>

Gupta, A. K. and D. K. Nagar (1999). Matrix Variate Distributions. Boca Raton: Chapman & Hall/CRC Press.

Examples

# Data Used
# if( !requireNamespace("datasets", quietly = TRUE)) { install.packages("datasets")} #part of baseR.
A <- datasets::CO2[1:10, 4:5]
M <- cbind(stats::rnorm(10, 435, 296), stats::rnorm(10, 27, 11))
V <- matrix(c(87, 13, 13, 112), nrow = 2, ncol = 2, byrow = TRUE)
V # Right covariance matrix (2 x 2), say the covariance between parameters.
U <- I(10) # Block of left-covariance matrix ( 84 x 84), say the covariance between subjects.

# PDF
dmatnorm(A, M, U, V)
dmatnorm(A, M, U, V, log = FALSE)

# Generating Probability Lower and Upper Bounds (They're matrices )
Lower <- matrix(rep(-1, 20), ncol = 2)
Upper <- matrix(rep(3, 20), ncol = 2)
Lower
Upper
# The probablity that a randomly chosen matrix A is between Lower and Upper
pmatnorm(Lower, Upper, M, U, V)

# CDF
pmatnorm(Lower = -Inf, Upper, M, U, V)
# entire domain = 1
pmatnorm(Lower = -Inf, Upper = Inf, M, U, V)

# Random generation
set.seed(123)
M <- cbind(rnorm(3, 435, 296), rnorm(3, 27, 11))
U <- diag(1, 3)
V <- matrix(c(10, 5, 5, 3), nrow = 2)
rmatnorm(1, M, U, V)

# M has a different sample size than U; will return an error.
## Not run: 
M <- cbind(rnorm(4, 435, 296), rnorm(4, 27, 11))
rmatnorm(M, U, V)

## End(Not run)

Generating Special Matrices

Description

Creates an Identity Matrix I and a Matrix of Ones J.

• I(): Creates an identity matrix where the number of columns is n. This is a diagonal matrix with all equal to one (1). An identity matrix is usually written as I. Names of rows and columns (dimnames) are included.

• J(): Creates a matrix of ones with any number of rows and columns. Names of rows and columns (dimnames) are included.

Usage

I(n)

J(n, m = n)

Arguments

See Also

Other matrix: tr(), vec()

Examples

# To create an identity matrix of order 12
I(2)
# To make a matrix of 6 rows and 10 columns of all ones
J(6, 10)
# To make a matrix of unity, dimensions 6 x 6.
J(6)

Matrix Trace

Description

Computes the trace of a square numeric matrix A.

Usage

tr(A)

Arguments

Note

If the argument is not a square numeric matrix, the function presents an error and terminates.

See Also

Other matrix: special.matrix, vec()

Examples

A <- matrix(seq(1, 16, 1), nrow = 4, byrow = TRUE)
A
tr(A)
tr(I(3))

Stacks a Matrix using matrix operator "vec"

Description

Returns a column vector that stacks the columns of A, a m x n matrix.

Usage

vec(A, use.Names = TRUE)

Arguments

Value

A vector with mn elements.

Note

1. Unlike other 'vec()' functions on CRAN, matrixNormal version inherits names from matrices to their vectorized forms.

2. vec() was adapted from Frederick Novomestky's matrixcalc. This function is edited so that it can take dimension names and return the matrix as a vector.

3. These functions were used as accessories for other matrixNormal functions.

References

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics. Second Edition, John Wiley, ed.

See Also

Other matrix: special.matrix, tr()

Examples

M <- matrix(c(4, 5, 6, 7, 8, 9), nrow = 3)
M
vec(M)
if (!requireNamespace("matrixcalc", quietly = TRUE)) {
# Compare vec from \pkg{matrixcalc} and new function.
matrixcalc::vec(M)
# The names are rownames(M):colnames(M) in that order.
# Very similar to matrixcalc but dimension names are different.
} else {
  message("you need to install the package matrixcalc to compare this example.")
}

Half-Vectorization of a matrix

Description

Stacks elements of the lower triangle of a numeric symmetric matrix A.

Usage

vech(A, use.Names = TRUE, tol = .Machine$double.eps^0.5)

Arguments

Details

For a symmetric matrix A, the vectorization of A contains more information than necessary. The half-vectorization, denoted vech(), of a symmetric square n by n matrix A is the vectorization of the lower triangular portion.

Value

A vector with n(n+1)/2 elements.

Note

Unlike other vech() functions available on CRAN, matrixNormal version may inherit names from matrices to their vectorized forms.

Examples

x <- matrix(c(1, 2, 2, 4),
  nrow = 2, byrow = TRUE,
  dimnames = list(1:2, c("Sex", "Smoker"))
)
print(x)

# Example 1
vech(x)
# If you just want the vectorized form
vech(x, use.Names = FALSE)

# Example 2: If one has NA's
x[1, 2] <- x[2, 1] <- NA
vech(x)