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inner
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:
inner(diag(7))
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) c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
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) c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
(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) c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
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]) c(LHS=LHS,RHS=RHS,diff=LHS-RHS)
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.
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)
Above we see that $\mathbf{x}^TM\mathbf{y} = -\mathbf{y}^TM\mathbf{x}$ where $M$ is antisymmetric.
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