The `inner()` function in the `stokes` package

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Spivak, 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:


Note how the rows of the tensor appear in arbitrary order. Verify:

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

Above, 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 <- quad.3form(M,x,y)
RHS <- f(V)

(function emulator::quad.3form() evaluates matrix products efficiently; quad.3form(M,x,y) returns $x^TMy$). Above we see agreement, to within numerical precision, of the dot product calculated two different ways: LHS uses quad.3form() 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 <- quad.3form(M1+M2,x,y)
RHS <- g(V)

Above 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])

Above 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

Above 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

Above we see that $\mathbf{x}^TM\mathbf{y} = -\mathbf{y}^TM\mathbf{x}$ where $M$ is antisymmetric.

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stokes documentation built on March 31, 2023, 11:58 p.m.