Inner product operator

Arguments

M

square matrix

Value

Returns a $k$-tensor, an inner product

Details

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.

Author

Robin K. S. Hankin

See also

Examples

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)