Calxsolver
Linear Algebra

Matrix Trace

The trace of a square matrix A, written tr(A), is the sum of the entries on its main diagonal. It equals the sum of A's eigenvalues and is a fundamental invariant under similarity transforms.

Matrix Trace

Sum of the diagonal entries of a square matrix.

Try:

Frequently asked questions

Why is the trace useful?

It equals the sum of the eigenvalues, is invariant under change of basis (tr(P⁻¹AP) = tr(A)) and shows up in many identities, e.g. tr(AB) = tr(BA).

Does the trace require a square matrix?

Yes. The trace is the sum of diagonal entries, so it is only defined when the number of rows equals the number of columns.

Is the trace linear?

Yes. tr(A + B) = tr(A) + tr(B) and tr(cA) = c·tr(A) for any scalar c.

Related solvers

Matrix Determinant Determinant of a square matrix. Matrix Multiplication Multiply two matrices A·B. Matrix Inverse Inverse of a square matrix via Gauss-Jordan. Matrix Addition & Subtraction Add or subtract two matrices entrywise. Vector Calculator Magnitude, dot product, cross product and angle. Matrix Transpose Swap rows and columns to get Aᵀ. Matrix Rank Number of linearly independent rows — by row reduction. Reduced Row Echelon Form (RREF) Gauss-Jordan elimination with every row operation shown. Solve a Linear System (Ax = b) Gauss-Jordan on the augmented matrix — unique, infinite or no solution. Eigenvalues and Eigenvectors Characteristic polynomial, real eigenvalues and one eigenvector per eigenvalue. Vector Projection Projection of a onto b, scalar projection and the perpendicular component.