trace

Sum of diagonal elements

Syntax

b = trace(A)

Description

b = trace(A) calculates the sum of the diagonal elements of matrix A:

$$\text{tr}\left(A\right)={\displaystyle \sum _{i=1}^n{a}_{ii}=a_{11}+a_{22}+\mathrm{...}+a_{nn}}.$$

Examples

Sum of Matrix Diagonal

Create a 3-by-3 matrix and calculate the sum of the diagonal elements.

A = [1 -5 2; 
    -3  7 9; 
     4 -1 6];

b = trace(A)

b = 14

The result $\mathrm{tr}\left(\mathit{A}\right)=14$ agrees with a manual calculation.

$$\mathit{A}=\left[\begin{array}{ccc}{\mathit{a}}_{11}& {\mathit{a}}_{12}& {\mathit{a}}_{13}\\ {\mathit{a}}_{21}& {\mathit{a}}_{22}& {\mathit{a}}_{23}\\ {\mathit{a}}_{31}& {\mathit{a}}_{32}& {\mathit{a}}_{33}\end{array}\right]=\left[\begin{array}{ccc}1& -5& 2\\ -3& 7& 9\\ 4& -1& 6\end{array}\right],$$

$$\mathrm{tr}\left(\mathit{A}\right)=\sum _{\mathit{i}=1}^3{\mathit{a}}_{\mathrm{ii}}={\mathit{a}}_{11}+{\mathit{a}}_{22}+{\mathit{a}}_{33}=1+7+6=14.$$

Matrix Trace Properties

Verify several properties of the trace of a matrix (up to round-off error).

Create two matrices. Verify that $\mathrm{tr}\left(\mathit{A}+\mathit{B}\right)=\mathrm{tr}\left(\mathit{A}\right)+\mathrm{tr}\left(\mathit{B}\right)$.

A = magic(3);
B = rand(3);
trace(A+B) 

ans = 17.4046

trace(A) + trace(B)

ans = 17.4046

Verify that $\mathrm{tr}\left(\mathit{A}\right)=\mathrm{tr}\left({\mathit{A}}^{\mathit{T})}\right)$.

trace(A)

ans = 15

trace(A')

ans = 15

Verify that $\mathrm{tr}\left({\mathit{A}}^{\mathit{T}}\mathit{B}\right)=\mathrm{tr}\left({\mathrm{AB}}^{\mathit{T}}\right)$.

trace(A'*B) 

ans = 22.1103

trace(A*B')

ans = 22.1103

Verify that $\mathrm{tr}\left(\mathrm{cA}\right)=\mathit{c}\mathrm{tr}\left(\mathit{A}\right)$ for a scalar $\mathit{c}$.

c = 5;
trace(c*A) 

ans = 75

c*trace(A)

ans = 75

Verify that the trace equals the sum of the eigenvalues $\mathrm{tr}\left(\mathit{A}\right)={\sum }_{\mathit{i}}{\lambda }_{\mathit{i}}$.

trace(A)

ans = 15

sum(eig(A))

ans = 15.0000

Input Arguments

A — Input matrix

square matrix

Input matrix, specified as a square matrix. A can be full or sparse.

Data Types: single | double
Complex Number Support: Yes

Algorithms

trace extracts the diagonal elements and adds them together with the command sum(diag(A)). The value of the trace is the same (up to round-off error) as the sum of the matrix eigenvalues sum(eig(A)).

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).