Trace of a Matrix
The trace of an n × n square matrix A is one of the simplest and most useful quantities in linear algebra. It is obtained by adding together the entries on the matrix's main diagonal.
In mathematical notation, the trace is defined as:
The trace is usually denoted by tr(A).
Example
Consider the following 3 × 3 matrix:
The entries on the main diagonal are 1, 5, and 9.
To find the trace, add these three values:
Therefore, the trace of matrix A is equal to 15.
Why Is the Trace Important?
The trace appears throughout linear algebra and its applications. It plays an important role in matrix theory, eigenvalue analysis, differential equations, statistics, quantum mechanics, and many other areas of mathematics and physics.
One particularly important result is that the trace of a matrix is equal to the sum of its eigenvalues, counted with their algebraic multiplicities.
Properties of the Trace
The trace satisfies several fundamental properties that make it a powerful mathematical tool.
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Scalar multiplication
Multiplying a matrix by a scalar multiplies its trace by the same scalar:
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Additivity
The trace of the sum of two matrices is equal to the sum of their traces:
Note. Together with the previous property, this shows that the trace is a linear functional on the vector space of square matrices.
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Invariance under transposition
A matrix and its transpose always have the same trace:
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Cyclic invariance
The trace of a product of matrices remains unchanged under cyclic permutations of the factors:
Non-cyclic permutations. The trace is generally not preserved under arbitrary rearrangements of the factors. For example, tr(ACB) and tr(CBA) are not necessarily equal to tr(ABC), because these products are not obtained through a cyclic permutation.
Although the trace is easy to compute, it captures important information about a matrix and appears in many theoretical and practical applications throughout mathematics.
