Documentation

trace

Sum of diagonal elements

Description

example

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

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

Examples

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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.

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

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

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

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 $\mathrm{tr}\left(\mathrm{cA}\right)=\mathit{c}\text{\hspace{0.17em}}\mathrm{tr}\left(\mathit{A}\right)$ 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

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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)).