`inner()`

function in the
`stokes`

package```
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:

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 = 1
```

Note how the rows of the tensor appear in arbitrary order, as per
`disordR`

dicipline (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 \(\mathbf{x}\cdot\mathbf{y}\) 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
## -3.410660e+00 -3.410660e+00 4.440892e-16
```

(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
## -5.418253e+00 -5.418253e+00 1.776357e-15
```

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)
```

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

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
```

`## [1] 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)
```

```
## LHS RHS diff
## 19.50013 19.50013 0.00000
```

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

Hankin, R. K. S. 2022a. “Disordered Vectors in R:
Introducing the

`disordR`

Package.” https://arxiv.org/abs/2210.03856; arXiv. https://doi.org/10.48550/ARXIV.2210.03856.
———. 2022b. “Stokes’s Theorem in R.” arXiv. https://doi.org/10.48550/ARXIV.2210.17008.

Spivak, M. 1965. *Calculus on Manifolds*. Addison-Wesley.