chol.tensor: Cholesky decomposition of a tensor

Description

A tensor can be seen as a linear mapping of a tensor to a tensor. This function computes its Cholesky decomposition.

Usage

chol.tensor(X,i,j,...,name="lambda")

Value

a tensor

Arguments

X: The tensor to be decomposed
i: The image dimensions of the linear mapping
j: The coimage dimensions of the linear mapping
name: The name of the eigenspace dimension. This is the dimension created by the decompositions, in which the eigenvectors are $e_{i}$
...: for generic use only

Author

K. Gerald van den Boogaart

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 $i_{1} . . . i_{d}$ .

chol.tensor: Computes for a tensor $a_{i_{1} \dots i_{d} j_{1} \dots j_{d}}$ representing a positive definit mapping form $R_{j}$ to $R_{i}$ with equal dimension structure in $i$ and $j$ its "Cholesky" decomposition $L_{i_{1} \dots i_{d} λ}$ such that $a_{i_{1} . . . i_{d} j_{1} . . . j_{d}} = \sum_{λ} L_{i_{1} . . . i_{d} λ} L_{j_{1} . . . j_{d} λ}$

Examples

Run this code



A <- to.tensor(rnorm(15),c(a=3,b=5))
AAt <- einstein.tensor(A,mark(A,i="a"))
ch <- chol.tensor(AAt,"a","a'",name="lambda")
#names(ch)[1]<-"lambda"
einstein.tensor(ch,mark(ch,i="a")) # AAt

A <- to.tensor(rnorm(30),c(a=3,b=5,c=2))
AAt <- einstein.tensor(A,mark(A,i="a"),by="c")
ch <- chol.tensor(AAt,"a","a'",name="lambda")
einstein.tensor(ch,mark(ch,i="a"),by="c") #AAt

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