finds the trace of the matrix or tensor list.
finds a generalized trace, combining terms with f instead of Plus.
goes down to level n in list.
- Tr[list] sums the diagonal elements list[[i,i,…]].
- Tr works for rectangular as well as square matrices and tensors.
- Tr can be used on SparseArray objects. »
Examplesopen allclose all
Basic Examples (1)
The trace of a matrix is the sum of the diagonal elements:
Trace of a numerical matrix:
Trace of a sparse matrix:
Generalizations & Extensions (6)
For a vector Tr gives the sum of the elements:
For a higher‐rank tensor, Tr gives the sum of elements with equal indices:
Apply a function to the diagonal elements of a matrix:
Extract the diagonal of a matrix as a list:
Only consider down to level 1; this adds the rows of the matrix:
Only consider down to level 2:
Find the determinant of a triangular matrix:
Define an inner product for the cone of positive definite matrices using :
Project the matrix onto the space spanned by the matrix :
Properties & Relations (3)
The trace of a matrix is invariant under similarity transformations:
The invariance means that the sum of the eigenvalues must equal the trace:
The Frobenius norm is defined as :
Tr[m,List] is equivalent to Diagonal[m] for a matrix m:
Introduced in 1999
Updated in 2003