sstvars (version 1.1.6)

diag_Omegas: Simultaneously diagonalize two covariance matrices

Description

diag_Omegas Simultaneously diagonalizes two covariance matrices using eigenvalue decomposition.

Usage

diag_Omegas(Omega1, Omega2)

Value

Returns a length \(d^2 + d\) vector where the first \(d^2\) elements are \(vec(W)\) with the columns of \(W\) being (specific) eigenvectors of the matrix \(\Omega_2\Omega_1^{-1}\) and the rest \(d\) elements are the corresponding eigenvalues "lambdas". The result satisfies \(WW' = Omega1\) and

\(Wdiag(lambdas)W' = Omega2\).

If Omega2 is not supplied, returns a vectorized symmetric (and pos. def.) square root matrix of Omega1.

Arguments

Omega1

a positive definite \((dxd)\) covariance matrix \((d>1)\)

Omega2

another positive definite \((dxd)\) covariance matrix

Warning

No argument checks! Does not work with dimension \(d=1\)!

Details

See the return value and Muirhead (1982), Theorem A9.9 for details.

References

• Muirhead R.J. 1982. Aspects of Multivariate Statistical Theory, Wiley.

Examples

# Create two (2x2) coviance matrices using the parameters W and lambdas:
d <- 2 # The dimension
W0 <- matrix(1:(d^2), nrow=2) # W
lambdas0 <- 1:d # The eigenvalues
(Omg1 <- W0%*%t(W0)) # The first covariance matrix
(Omg2 <- W0%*%diag(lambdas0)%*%t(W0)) # The second covariance matrix

# Then simultaneously diagonalize the covariance matrices:
res <- diag_Omegas(Omg1, Omg2)

# Recover W:
W <- matrix(res[1:(d^2)], nrow=d, byrow=FALSE)
tcrossprod(W) # == Omg1, the first covariance matrix

# Recover lambdas:
lambdas <- res[(d^2 + 1):(d^2 + d)]
W%*%diag(lambdas)%*%t(W) # == Omg2, the second covariance matrix