Difference between revisions of "Trace"
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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>. |
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| + | == Properties == | ||
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| + | 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> and any scalar <math>c</math>. | ||
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| + | == See also == | ||
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| + | * [[Characteristic polynomial]] | ||
[[Category:Linear algebra]] | [[Category:Linear algebra]] | ||
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| + | {{stub}} | ||
Latest revision as of 18:27, 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 
and any scalar 
.
See also
This article is a stub. Help us out by expanding it.
