Matrix Trace

From GM-RKB
(Redirected from matrix trace)
Jump to navigation Jump to search

A Matrix Trace is a sum of the diagonal elements of square matrix.

  • Context:
    • It is defined as [math]\displaystyle{ \operatorname{tr}(A) =\sum_{i=1}^{n} a_{ii} }[/math], where A is square matrix [math]\displaystyle{ n\times n }[/math]
  • Example(s):
    • [math]\displaystyle{ \text{if}\;A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \text{then the trace of A is } \operatorname{tr}(A) = 1+5+9=15 }[/math] .
  • Counter-Example(s):
  • See: Frobenius_Norm, Matrix Norm, Square Matrix.


References

2015

[math]\displaystyle{ \operatorname{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{ii} }[/math]
where ann denotes the entry on the n-th row and n-th column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. Note that the trace is only defined for a square matrix (i.e., n × n).
The trace is related to the derivative of the determinant (see Jacobi's formula).
The term trace is a calque from the German Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "tr".

1999

[math]\displaystyle{ Tr(A)=\sum_{i=1}^na_{ii} }[/math]
i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr[list]. In group theory, traces are known as "group characters."