Matrix Trace Calculator
Calculate the trace of any square matrix (sum of diagonal elements). Enter your matrix below and get instant results with detailed step-by-step calculations and mathematical properties.
Matrix A
Diagonal Elements
Understanding Matrix Trace
Simple Definition
Trace is the sum of all diagonal elements: tr(A) = a₁₁ + a₂₂ + ... + aₙₙ
Square Matrices Only
Trace is only defined for square matrices (n×n). Non-square matrices don't have a trace.
Eigenvalue Connection
The trace equals the sum of all eigenvalues (counting multiplicities).
Linear Property
tr(A + B) = tr(A) + tr(B) and tr(cA) = c·tr(A) for scalar c.
Mathematical Theory & Properties
What is Matrix Trace?
The trace of a square matrix A, denoted tr(A) or Tr(A), is the sum of the elements on the main diagonal. It's one of the most fundamental invariants in linear algebra, providing important information about the matrix's properties and the linear transformation it represents.
Definition: tr(A) = Σᵢ aᵢᵢ = a₁₁ + a₂₂ + ... + aₙₙ
Eigenvalue relation: tr(A) = λ₁ + λ₂ + ... + λₙ
Where λᵢ are the eigenvalues of A
Historical Background
The concept of matrix trace emerged naturally from the study of linear transformations and quadratic forms. While the formal definition developed alongside matrix theory in the 19th century, the underlying concept can be traced back to earlier work on determinants and linear equations.
The trace gained particular importance with the development of eigenvalue theory by Augustin-Louis Cauchy and later mathematicians, who established its connection to the sum of eigenvalues, making it a crucial tool in spectral analysis and matrix theory.
Properties of Matrix Trace
Linearity
tr(A + B) = tr(A) + tr(B)
Scalar Multiplication
tr(cA) = c·tr(A)
Cyclic Property
tr(AB) = tr(BA)
Transpose Invariance
tr(A) = tr(AT)
Similarity Invariance
tr(P⁻¹AP) = tr(A)
Zero Matrix
tr(0) = 0
Real-World Applications
Physics & Engineering
Calculating total energy, momentum conservation, and system invariants
Statistics
Sum of variances in covariance matrices and principal component analysis
Machine Learning
Regularization terms, neural network analysis, and optimization algorithms
Computer Graphics
Transformation analysis, scaling factors, and geometric invariants
Frequently Asked Questions
No, the trace is only defined for square matrices (n×n). Non-square matrices don't have a main diagonal in the traditional sense, so the trace concept doesn't apply to them.
The trace of a matrix equals the sum of all its eigenvalues (counting multiplicities). This is a fundamental result: tr(A) = λ₁ + λ₂ + ... + λₙ, where λᵢ are the eigenvalues of A.
When tr(A) = 0, the sum of eigenvalues is zero. This doesn't mean the matrix is singular, but it indicates that positive and negative eigenvalues balance out. Such matrices are important in physics and optimization.
The cyclic property shows that trace is invariant under cyclic permutations. This is crucial in quantum mechanics, where it ensures physical observables are independent of basis choice, and in similarity transformations.
Trace appears in regularization terms (trace norm), covariance matrix analysis, neural network weight analysis, and optimization algorithms. It's particularly important in dimensionality reduction and matrix factorization techniques.
While both are matrix invariants, they measure different properties. Trace is the sum of eigenvalues (linear), while determinant is their product (multiplicative). Both are preserved under similarity transformations.
Yes, the trace can be any real number (or complex for complex matrices). A negative trace means the sum of diagonal elements is negative, which often occurs in systems with dominant negative eigenvalues.