The definition of the RV-coefficient makes use of ideas^[5] concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued random variables. Note that standard usage is to have matrices for the variances and covariances of vector random variables. Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.

Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by

${\displaystyle \Sigma _{XY}=\operatorname {E} (XY^{\top })\,,}$ 

then the scalar-valued covariance (denoted by COVV) is defined by^[5]

${\displaystyle \operatorname {COVV} (X,Y)=\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})\,.}$ 

The scalar-valued variance is defined correspondingly:

${\displaystyle \operatorname {VAV} (X)=\operatorname {Tr} (\Sigma _{XX}^{2})\,.}$ 

With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.^[5]

Then the RV-coefficient is defined by^[5]

${\displaystyle \mathrm {RV} (X,Y)={\frac {\operatorname {COVV} (X,Y)}{\sqrt {\operatorname {VAV} (X)\operatorname {VAV} (Y)}}}\,.}$ 

Even though the coefficient takes values between 0 and 1 by construction, it seldom attains values close to 1 as the denominator is often too large with respect to the maximal attainable value of the denominator.^[6]

Given known diagonal blocks ${\displaystyle \Sigma _{XX}}$  and ${\displaystyle \Sigma _{YY}}$  of dimensions ${\displaystyle p\times p}$  and ${\displaystyle q\times q}$  respectively, assuming that ${\displaystyle p\leq q}$  without loss of generality, it has been proved^[7] that the maximal attainable numerator is ${\displaystyle \operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y}),}$  where ${\displaystyle \Lambda _{X}}$  (resp. ${\displaystyle \Lambda _{Y}}$ ) denotes the diagonal matrix of the eigenvalues of ${\displaystyle \Sigma _{XX}}$ (resp. ${\displaystyle \Sigma _{YY}}$ ) sorted decreasingly from the upper leftmost corner to the lower rightmost corner and ${\displaystyle \Pi }$  is the ${\displaystyle p\times q}$  matrix ${\displaystyle (I_{p}\ 0_{p\times (q-p)})}$ .

In light of this, Mordant and Segers^[7] proposed an adjusted version of the RV coefficient in which the denominator is the maximal value attainable by the numerator. It reads

${\displaystyle {\bar {\operatorname {RV} }}(X,Y)={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\operatorname {Tr} (\Lambda _{X}\Pi \Lambda _{Y})}}={\frac {\operatorname {Tr} (\Sigma _{XY}\Sigma _{YX})}{\sum _{j=1}^{min(p,q)}(\Lambda _{X})_{j,j}(\Lambda _{Y})_{j,j}}}.}$ 

The impact of this adjustment is clearly visible in practice.^[7]

• Congruence coefficient
• Distance correlation