Let we have Track and Vertex(hit), and we want to update track by vertex(hit). How to do it?

 Track T has 5 parameters and vertex V has 3 ones.

 Find the point on track nearest to vertex (DCA point). Set coordinate system where zero in DCA point,
Z_axis along the track, Y_axis coming thru vertex V and X_axis orthogonal to them.

In this system Track has parameters T(x,y,Pt,Theta,Phi), vertex (x₀,y₀,z₀=0)
Or T = (X , P) where X = (x,y) and P = (Pt,Theta,Phi)

vertex  V(x₀,y₀) == V(X₀)
T₀ = (0,P₀)
V₀ =( X₀)

T_F = (X_F, P_F)
V_F = (X_F)

                                          | g_xx    g_xp |
Error matrix G_xp for track =   |               |
                                          | g_xp   g_pp |

                          | gi_xx  gi_xp |
 ( GI_xp)^-1 = GI =   |              |                             
                          | gi_xp gi_pp |

Error matrix  for V is q_xx 
( q_xx)^-1  = qi

Case 1: V is a hit, i.e. measurement of point on the track. We want to estimate new track parameters T₁
             closest to both T₀ and V in the sense of Xi²

a)  Xi²   = (T_F-T₀) * (GI) * (T_F-T₀) +  (V_F  - V₀) * qi * (V_F  - V₀)

                               | qi_xx  0 |
Let introduce: QI =   |           | 
                               |  0    0 |
and T_v = T(X₀ ,0)

Then we can rewrite expression a) as
b)  Xi²   = (T_F-T₀) * GI * (T_F-T₀) +  (T_F  - T_v) * QI * (T_F  - T_v )

Assigning all derivatives to zero we got:

c)   GI * (T_F-T₀) +   QI * (T_F  - T_v ) = 0
     ( GI +QI)*T_F = GI * T₀ +   QI *T_v
 
d)     T_F = ( GI +QI) ^{-1 *} ( GI * T₀ +   QI *T_v)

     and ( GI +QI) ^-1 == error matrix for   T_F

Case 2: Vertex is not a hit but real  vertex. That means T_F must pass thru vertex.
            Then T_F = T(X₀,P_F)

e)  Xi²   = (T_F-T₀) * (GI) * (T_F-T₀)

    (T_F-T₀) == T(X₀,P_F) - T( 0,P₀ ) == T(X₀,D_F)                          where D_F = P_F -P₀
f)  Xi²   = (   g_ixx*X₀ +  gi_xp * D_F  )   * X₀ + (   g_ixp*X₀ +  gi_{pp *} D_F  ) * D_F

Assigning all derivatives to zero we got:
g)                  gi_xp  * X₀ +  gi_{pp *} D_F  = 0
h)                 D_F  = -  (gi_pp^)-1   * gi_xp  * X₀
    and (gi_pp^)-1  the error matrix of D_F

You see the re is no dependency from Q or QI
Also from h) D_F  linearly depends of  X₀
Then at the end error matrix of D_F = (gi_pp^)-1   * gi_xp ) * q_xx * (gi_pp^)-1   * gi_xp )