power.tensor {tensorA}

Compute the power of a symmetric tensor

Description

A tensor can be seen as a linear mapping of a tensor to a tensor. If domain and image are the same and the tensor is definite, we can define powers.

Usage

power.tensor(X,i,j,p=0.5,by=NULL)

Arguments

X

The tensor to be decomposed

i

The image dimensions of the linear mapping

j

The domain dimensions of the linear mapping

p

the power of the tensor to be computed

by

the operation is done in parallel for these dimensions

Details

A tensor can be seen as a linear mapping of a tensor to a tensor. Let denote R_i the space of real tensors with dimensions i_1...i_d.
To compute a power dim(X)[i] and dim(X)[j] need to be equal and the tensor symmetric between these dimension. Some exponents are only valid with positive definite mappings. None of these conditions is checked.

Value

a tensor

Note

symmetry of the matrix is assumed but not checked.

Author(s)

K. Gerald van den Boogaart

See Also

svd.tensor,

Examples

A <- to.tensor(rnorm(120),c(a=2,b=2,c=5,d=3,e=2))
AAt <- A %e% mark(A,"'",c("a","b"))
AAt

power.tensor(AAt,c("a","b"),c("a'","b'"),-1)

inv.tensor(AAt,c("a","b"))

power.tensor(AAt,c("a","b"),c("a'","b'"),2)
mul.tensor(AAt,c("a","b"),AAt,c("a'","b'"))

power.tensor(power.tensor(AAt,c("a","b"),c("a'","b'"),1/pi),
                     c("a","b"),c("a'","b'"),pi)


AAt <- einstein.tensor(A , mark(A,"'",c("a","b")),by="e")

power.tensor(AAt,c("a","b"),c("a'","b'"),-1,by="e")

inv.tensor(AAt,c("a","b"),by="e")

power.tensor(AAt,c("a","b"),c("a'","b'"),2,by="e")
mul.tensor(AAt,c("a","b"),AAt,c("a'","b'"),by="e")

power.tensor(power.tensor(AAt,c("a","b"),c("a'","b'"),1/pi,by="e"),
c("a","b"),c("a'","b'"),pi,by="e")