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The '''trace''' of a square <math>n \times n</math> [[matrix]] is the sum of the elements of the main diagonal of the matrix. | The '''trace''' of a square <math>n \times n</math> [[matrix]] is the sum of the elements of the main diagonal of the matrix. For example, the trace of the matrix <math>M = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}</math> is <math>\text{tr}(M) = 1 + 5 + 9 = 15</math>. | ||
== Properties == | |||
Viewed as a function from <math>n \times n</math> matrices to the underlying [[field]] (frequently the [[real number]]s), the trace is a [[linear map]]: it is not difficult to verify that <math>\text{tr}(cA + B) = c\text{tr}(A) + \text{tr}(B)</math> for any <math>n \times n</math> matrices <math>A</math> and <math>B</math>. | |||
== See also == | |||
* [[Characteristic polynomial]] | |||
[[Category:Linear algebra]] | [[Category:Linear algebra]] | ||
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Revision as of 18:26, 2 March 2010
The trace of a square 
matrix is the sum of the elements of the main diagonal of the matrix. For example, the trace of the matrix 
is 
.
Properties
Viewed as a function from matrices to the underlying field (frequently the real numbers), the trace is a linear map: it is not difficult to verify that 
for any matrices 
and 
.
See also
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