inner.RdThe inner product
inner(M)Returns a \(k\)-tensor, an inner product
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 \(\mathbf{x}^TM\mathbf{y}\) where \(M\) is a matrix. 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.
Given a square matrix \(M\), function inner(M) returns the
\(2\)-form that maps \(\mathbf{x},\mathbf{y}\) to
\(\mathbf{x}^TM\mathbf{y}\). Non-square matrices are
effectively padded with zeros.
A short vignette is provided with the package: type
vignette("inner") at the commandline.
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
inner(matrix(1:9,3,3))
#> A linear map from V^2 to R with V=R^3:
#> val
#> 3 3 = 9
#> 2 3 = 8
#> 1 3 = 7
#> 3 2 = 6
#> 2 2 = 5
#> 1 2 = 4
#> 3 1 = 3
#> 2 1 = 2
#> 1 1 = 1
## Compare the following two:
Alt(inner(matrix(1:9,3,3))) # An alternating k tensor
#> A linear map from V^2 to R with V=R^3:
#> val
#> 3 2 = -1
#> 3 1 = -2
#> 2 3 = 1
#> 1 3 = 2
#> 2 1 = -1
#> 1 2 = 1
as.kform(inner(matrix(1:9,3,3))) # Same thing coerced to a kform
#> An alternating linear map from V^2 to R with V=R^3:
#> val
#> 2 3 = 2
#> 1 3 = 4
#> 1 2 = 2
f <- as.function(inner(diag(7)))
X <- matrix(rnorm(14),ncol=2) # random element of (R^7)^2
f(X) - sum(X[,1]*X[,2]) # zero to numerical precision
#> [1] 0
## verify positive-definiteness:
g <- as.function(inner(crossprod(matrix(rnorm(56),8,7))))
stopifnot(g(kronecker(rnorm(7),t(c(1,1))))>0)