Trace¶

Definition¶

The trace of a square matrix $\boldsymbol{A}$ is the sum of its diagonal entries: $\displaystyle \operatorname{tr}(\boldsymbol{A})=\sum_{i=1}^n a_{ii}$ $i = 1 \sum n a_{ii}$ .

Interpretation¶

The trace is a simple scalar summary of the main diagonal: add the diagonal entries and ignore the rest. Although it is easy to compute, it carries information about the eigenvalues.

For every square matrix, the trace is the sum of the complex eigenvalues, counted with algebraic multiplicity, and the determinant is their product. For a $2\times2$ matrix this can be seen in the characteristic polynomial $\lambda^2-\operatorname{tr}(\boldsymbol{A})\lambda+\det\boldsymbol{A}$ $det A$ . For $\begin{pmatrix}1&2\\3&4\end{pmatrix}$ the eigenvalues are $\tfrac{1}{2}(5\pm\sqrt{33})$ $33)$ , yet their sum $5$ and product $-2$ need no computation.

Change the entries of $\boldsymbol{A}$ : the eigenvalues always add up to $\operatorname{tr}(\boldsymbol{A})$ and multiply to $\det(\boldsymbol{A})$ , so the parabola is symmetric about $\operatorname{tr}(\boldsymbol{A})/2$ .

The trace is linear, and $\operatorname{tr}(\boldsymbol{A}\boldsymbol{B})=\operatorname{tr}(\boldsymbol{B}\boldsymbol{A})$ even when $\boldsymbol{A}\boldsymbol{B}\neq\boldsymbol{B}\boldsymbol{A}$ $B A$ ; hence $\operatorname{tr}(\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P})=\operatorname{tr}(\boldsymbol{A})$ $tr (A)$ , so the trace does not change under diagonalization. It is not multiplicative: for $\boldsymbol{A}=\boldsymbol{B}=\boldsymbol{I}_2$ $I_{2}$ , $\operatorname{tr}(\boldsymbol{A}\boldsymbol{B})=2$ $2$ , while $\operatorname{tr}(\boldsymbol{A})\operatorname{tr}(\boldsymbol{B})=4$ $4$ .

Key Results¶

The sum of the complex eigenvalues of a square matrix equals its trace, counted with algebraic multiplicity.
The product of the complex eigenvalues equals the determinant, counted with algebraic multiplicity.

Examples¶

$\operatorname{tr}\begin{pmatrix}1&2\\3&4\end{pmatrix}=5$ $5$ .
$\operatorname{tr}(\boldsymbol{I}_n)=n$ $n$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 6

Aliases: matrix trace Symbols: lambda, square root, sum

Last updated: 20 October 2026 03:02