Matrix Trace

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):
  • Matrix Norm.
• See: Frobenius_Norm, Matrix Norm, Square Matrix.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Trace_(linear_algebra)
  • QUOTE: In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

[math]\displaystyle{ \operatorname{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}=\sum_{i=1}^{n} a_{ii} }[/math]

where a_nn 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

• (Wolfram Mathworld , 1999) ⇒ http://mathworld.wolfram.com/MatrixTrace.html
  • QUOTE: The trace of an n×n square matrix A is defined to be

[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."

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