Trace of a Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Definition: The trace of a square matrix A, denoted by tr(A), is the sum of its main diagonal entries. (Handout 2, Page 3)

◇Key Concepts

Example Properties of the Trace

◇Details / Theory

Definition: The trace of a square matrix A, denoted by tr(A), is the sum of its main diagonal entries. (Handout 2, Page 3)
tr(A)=i=1∑n​aii​=a11​+a22​+⋯+ann​ Example: For A=[2​1​0 −1​4​2 10​5​3​], tr(A)=2+4+3=9. Properties of the Trace: If A and B are n×n matrices and c is a scalar: (Handout 2, Page 3 for first two; others are common properties)

1. tr(A+B)=tr(A)+tr(B)

2. tr(AB)=tr(BA)
   Personal Note: This is a very useful property, especially since AB=BA in general.

3. tr(cA)=c⋅tr(A)

4. tr(A⊤)=tr(A)

◇Common Pitfalls

Trace is only defined for square matrices. $\text{tr}(AB) = \text{tr}(BA)$ but $\text{tr}(AB) \neq \text{tr}(A) \cdot \text{tr}(B)$ in general.

◇Summary

TL;DR — $\text{tr}(A) = \sum a_{ii}$ (sum of diagonal entries). Linear, cyclic ($\text{tr}(AB) = \text{tr}(BA)$), and equals the sum of eigenvalues.

◇Links

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◇References & Navigation

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• Related Notes: ◇Skew-Symmetric Matrix, ◇Upper Triangular Matrix, ◇Matrices Detailed