inner

function (M) 
{
    ktensor(spray(expand.grid(seq_len(nrow(M)), seq_len(ncol(M))), 
        c(M)))
}

To cite the stokes package in publications, please use Hankin (2022b). Spivak (1965), in a memorable passage, states:

Function inner() returns the inner product corresponding to a matrix M. Spivak’s definition requires M to be positive-definite, but that is not necessary in the package. The inner product of two vectors x and y is usually written ⟨x, y⟩ or x ⋅ y, but the most general form would be x^TMy. Noting that inner products are multilinear, that is ⟨x, ay + bz⟩ = a⟨x, y⟩ + b⟨x, z⟩ and ⟨ax + by, z⟩ = a⟨x, z⟩ + b⟨y, z⟩ we see that the inner product is indeed a multilinear map, that is, a tensor.

We can start with the simplest inner product, the identity matrix:

inner(diag(7))

## A linear map from V^2 to R with V=R^7:
##          val
##  6 6  =    1
##  7 7  =    1
##  5 5  =    1
##  3 3  =    1
##  2 2  =    1
##  4 4  =    1
##  1 1  =    1

Note how the rows of the tensor appear in arbitrary order, as per disordR discipline (Hankin 2022a). Verify:

x <- rnorm(7)
y <- rnorm(7)
V <- cbind(x,y)
LHS <- sum(x*y)
RHS <- as.function(inner(diag(7)))(V)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

##      LHS      RHS     diff 
## 5.503805 5.503805 0.000000

Above, we see agreement between x ⋅ y calculated directly [LHS] and using inner() [RHS]. A more stringent test would be to use a general matrix:

M <- matrix(rnorm(49),7,7)
f <- as.function(inner(M))
LHS <- quad3.form(M,x,y)
RHS <- f(V)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

##      LHS      RHS     diff 
## -3.41066 -3.41066  0.00000

(function quadform::quad3.form() evaluates matrix products efficiently; quad3.form(M,x,y) returns x^TMy). Above we see agreement, to within numerical precision, of the dot product calculated two different ways: LHS uses quad3.form() and RHS uses inner(). Of course, we would expect inner() to be a homomorphism:

M1 <- matrix(rnorm(49),7,7)
M2 <- matrix(rnorm(49),7,7)
g <- as.function(inner(M1+M2))
LHS <- quad3.form(M1+M2,x,y)
RHS <- g(V)
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

##       LHS       RHS      diff 
## -5.418253 -5.418253  0.000000

Above we see numerical verification of the fact that I(M₁ + M₂) = I(M₁) + I(M₂), by evaluation at x, y, again with LHS using direct matrix algebra and RHS using inner(). Now, if the matrix is symmetric the corresponding inner product should also be symmetric:

h <- as.function(inner(M1 + t(M1))) # send inner() a symmetric matrix
LHS <- h(V)
RHS <- h(V[,2:1])
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)

##       LHS       RHS      diff 
## -22.52436 -22.52436   0.00000

Above we see that x^TMy = y^TMx. Further, a positive-definite matrix should return a positive quadratic form:

M3 <- crossprod(matrix(rnorm(56),8,7))  # 7x7 pos-def matrix
as.function(inner(M3))(kronecker(rnorm(7),t(c(1,1))))>0  # should be TRUE

## [1] TRUE

Above we see the second line evaluating x^TMx with M positive-definite, and correctly returning a non-negative value.

jj <- matrix(rpois(49,lambda=3.2),7,7)
M <- jj-t(jj) # M is antisymmetric
f <- as.function(inner(M))
LHS <- f(V)
RHS <- -f(V[,2:1])   # NB negative as we are checking for an alternating form
c(LHS=LHS,RHS=RHS,diff=LHS-RHS) 

##      LHS      RHS     diff 
## 19.50013 19.50013  0.00000