innerfunction (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:
The reader is already familiar with certain tensors, aside from members of V^*. The first example is the inner product \left\langle{,}\right\rangle\in{\mathcal J}^2(\mathbb{R}^n). On the grounds that any good mathematical commodity is worth generalizing, we define an inner product on V to be a 2-tensor T such that T is symmetric, that is T(v,w)=T(w,v) for v,w\in V and such that T is positive-definite, that is, T(v,v) > 0 if v\neq 0. We distinguish \left\langle{,}\right\rangle as the usual inner product on \mathbb{R}^n.
- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 77
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
\mathbf{x} and \mathbf{y} is usually written \left\langle\mathbf{x},\mathbf{y}\right\rangle
or \mathbf{x}\cdot\mathbf{y}, but the
most general form would be \mathbf{x}^TM\mathbf{y}. Noting that inner
products are multilinear, that is \left\langle\mathbf{x},a\mathbf{y}+b\mathbf{z}\right\rangle=a\left\langle\mathbf{x},\mathbf{y}\right\rangle
+ b\left\langle\mathbf{x},\mathbf{z}\right\rangle and \left\langle
a\mathbf{x} +
b\mathbf{y},\mathbf{z}\right\rangle=a\left\langle\mathbf{x},\mathbf{z}\right\rangle
+ b\left\langle\mathbf{y},\mathbf{z}\right\rangle 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:
## 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 = 1Note 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 0.000000Above, we see agreement between \mathbf{x}\cdot\mathbf{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 0.000000Above we see numerical verification of the fact that I(M_1+M_2)=I(M_1)+I(M_2), by evaluation at
\mathbf{x},\mathbf{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 0.00000Above we see that \mathbf{x}^TM\mathbf{y} = \mathbf{y}^TM\mathbf{x}. 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] TRUEAbove we see the second line evaluating \mathbf{x}^TM\mathbf{x} with M positive-definite, and correctly returning a non-negative value.
Alternating forms
The inner product on an antisymmetric matrix should be alternating:
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 0.00000Above we see that \mathbf{x}^TM\mathbf{y} = -\mathbf{y}^TM\mathbf{x} where M is antisymmetric.
References
disordR Package.” https://arxiv.org/abs/2210.03856; arXiv. https://doi.org/10.48550/ARXIV.2210.03856.