// Start with identity
changeOfBasis = matrix.identity

// Construct the basis which represents the difference between the two
// spaces. For a typical left/right handed space, the forward vector is
// inverted. Note that this is not well defined though, as there are many
// possible bases for left and right handed systems.
changeOfBasis[2,2] = -1;

// Apply the similarity transform.
newValue = changeOfBasis * value * changeOfBasis.inverse

// Note that in this case, the inverse is the change of basis, so this can
// be simplified to:
newValue = changeOfBasis * value * changeOfBasis

// Which can be simplified further to 6 multiplies by -1.