Why is the trace of a matrix important?

Question

Lower division linear algebra course at my university taught the simple computation steps to finding a trace of a matrix, but not the intuition nor the purpose for it. What information is gained from summing the diagonal entries?

Why is the trace of a matrix important?

Answer 1

The trace has useful for multiple reasons. Besides the fact that it is an invariant like the determinant, it allows use to generalize several interesting operations to more general cases.

1. It is the sum of the eigenvalues and this is already an important property that can be exploited for proving certain results.
2. It has a lot of nice properties such as linearity, invariance by transposition and basis change, and perhaps more importantly invariance by cyclic permutations; i.e. $\mathrm{trace}(ABC)=\mathrm{trace}(CAB)=\mathrm{trace}(BCA)$ for square matrices $A,B,C$. Again those properties are extremely convenient to have.
3. The inner-product on $\mathbb{R}$, $\langle x,y\rangle=x^Ty$ is generalized to matrices in $\mathbb{R}^{n\times n}$ as $\langle X,Y\rangle=\mathrm{trace}(X^TY)$ and the same use can be done with matrices as with vectors, so we can talk of projections, induced-norms, distances, operators and their adjoints, etc. The norm induced by this inner-product is the so-called Frobenius norm, which coincides with the sum of the singular values of the matrix. Such an inner product has important application in optimization and, especially, in semidefinite programming. It also has applications in the analysis of matrix-valued dynamical systems where the trace allows us to build suitable Lyapunov functions.
4. This also generalizes the inner product on infinite-dimensional space such as $\ell_2[0,\infty)$ and $L_2[0,\infty)$ spaces where $$\langle x,y\rangle_{\ell_2}=\sum_{i=0}^\infty x_i^*y_i\quad \mathrm{and}\quad \langle x,y\rangle_{L_2}=\int_0^\infty x(s)^*y(s)ds,$$

to the matrix case as

$$\langle X,Y\rangle_{\ell_2}=\mathrm{trace}\left(\sum_{i=0}^\infty X_i^*Y_i\right)\quad \mathrm{and}\quad \langle X,Y\rangle_{L_2}=\mathrm{trace}\left(\int_0^\infty Y(s)^*X(s)ds\right),$$

and this propagates all the results and concepts to the matrix case with no effort whatsoever. This also has applications in the analysis of matrix-valued dynamical systems.