Square root of a matrix
For a matrix A its square root is such a matrix B which satisfies A = B B.
Computation is done using spectral decomposition. When calculating the square roots of eigenvalues, always a root with positive real part and a sign of the imaginary part the same as the sign of the imaginary eigenvalue part is taken.
MatSqrt(A)
A |
either a numeric vector in which case square roots of each element of A is returned or a squared matrix. |
Either a numeric vector or a matrix.
Arnošt Komárek arnost.komarek[AT]mff.cuni.cz
MatSqrt(0:4)
MatSqrt((-4):0)
MatSqrt(c(-1, 1, -2, 2))
A <- (1:4) %*% t(1:4)
sqrtA <- MatSqrt(A)
sqrtA
round(sqrtA %*% sqrtA - A, 13)
### The following example crashes on r-devel Windows x64 x86_64,
### on r-patched Linux x86_64
### due to failure of LAPACK zgesv routine
###
### Commented on 16/01/2010
###
# B <- -A
# sqrtB <- MatSqrt(B)
# sqrtB
# round(Re(sqrtB %*% sqrtB - B), 13)
# round(Im(sqrtB %*% sqrtB - B), 13)
V <- eigen(A)$vectors
sqrtV <- MatSqrt(V)
sqrtV
round(sqrtV %*% sqrtV - V, 14)
Sigma <- matrix(c(1, 1, 1.5, 1, 4, 4.2, 1.5, 4.2, 9), nrow=3)
sqrtSigma <- MatSqrt(Sigma)
sqrtSigma
round(sqrtSigma %*% sqrtSigma - Sigma, 13)
D4 <- matrix(c(5, -4, 1, 0, 0,
-4, 6, -4, 1, 0,
1, -4, 6, -4, 1,
0, 1, -4, 6, -4,
0, 0, 1, -4, 5), nrow=5)
sqrtD4 <- MatSqrt(D4)
sqrtD4[abs(sqrtD4) < 1e-15] <- 0
sqrtD4
round(sqrtD4 %*% sqrtD4 - D4, 14)
X <- matrix(c(7, 15, 10, 22), nrow=2)
sqrtX <- MatSqrt(X)
sqrtX %*% sqrtX - XPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.