The trace of an matrix is the sum of its diagonal elements: . The trace is linear, that is, , and .

A key fact is that the trace is also the sum of the eigenvalues. The proof is by considering the characteristic polynomial . The roots of are the eigenvalues of , so can be factorized

and so . The Laplace expansion of shows that the coefficient of is . Equating these two expressions for gives

A consequence of (1) is that any transformation that preserves the eigenvalues preserves the trace. Therefore the trace is unchanged under similarity transformations: for any nonsingular .

An an example of how the trace can be useful, suppose is a symmetric and orthogonal matrix, so that its eigenvalues are . If there are eigenvalues and eigenvalues then and . Therefore and .

Another important property is that for an matrix and an matrix ,

(despite the fact that in general). The proof is simple:

This simple fact can have non-obvious consequences. For example, consider the equation in matrices. Taking the trace gives , which is a contradiction. Therefore the equation has no solution.

The relation (2) gives for matrices , , and , that is,

So we can cyclically permute terms in a matrix product without changing the trace.

As an example of the use of (2) and (3), if and are -vectors then . If is an matrix then can be evaluated without forming the matrix since, by (3), .

The trace is useful in calculations with the Frobenius norm of an matrix:

where denotes the conjugate transpose. For example, we can generalize the formula for a complex number to an matrix by splitting into its Hermitian and skew-Hermitian parts:

where and . Then

If a matrix is not explicitly known but we can compute matrix–vector products with it then the trace can be estimated by

where the vector has elements independently drawn from the standard normal distribution with mean and variance . The expectation of this estimate is

since for and for all , so . This stochastic estimate, which is due to Hutchinson, is therefore unbiased.

## References

- Haim Avron and Sivan Toledo, Randomized Algorithms for Estimating the Trace of an Implicit Symmetric Positive Semi-definite Matrix, J. ACM 58, 8:1-8:34, 2011.

## Related Blog Posts

- What Is a Matrix Norm? (2021)
- What Is an Eigenvalue? (2022)

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.