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Trace of a matrix

by , PhD

The trace of a square matrix is the sum of its diagonal elements.

The trace enjoys several properties that are often very useful when proving results in matrix algebra and its applications.

Table of Contents

Definition

Let us start with a formal definition.

Definition Let A be a $K	imes K$ matrix. Then, its trace, denoted by [eq1] or [eq2], is the sum of its diagonal elements:[eq3]

Examples

Some examples follow.

Example Define the matrix[eq4]Then, its trace is[eq5]

Example Define the matrix[eq6]Then, its trace is[eq7]

Properties

The following subsections report some useful properties of the trace operator.

Trace of a sum

The trace of a sum of two matrices is equal to the sum of their trace.

Proposition Let A and $B$ be two $K	imes K$ matrices. Then, [eq8]

Proof

Remember that the sum of two matrices is performed by summing each element of one matrix to the corresponding element of the other matrix (see the lecture on Matrix addition). As a consequence,[eq9]

Trace of a scalar multiple

The next proposition tells us what happens to the trace when a matrix is multiplied by a scalar.

Proposition Let A be a $K	imes K$ matrix and $lpha $ a scalar. Then,[eq10]

Proof

Remember that the multiplication of a matrix by a scalar is performed by multiplying each entry of the matrix by the given scalar (see the lecture on Multiplication of a matrix by a scalar). As a consequence,[eq11]

Trace of a linear combination

The two properties above (trace of sums and scalar multiples) imply that the trace of a linear combination is equal to the linear combination of the traces.

Proposition Let A and $B$ be two $K	imes K$ matrices and $lpha $ and $eta $ two scalars. Then, [eq12]

Trace of the transpose of a matrix

Transposing a matrix does not change its trace.

Proposition Let A be a $K	imes K$ matrix. Then,[eq13]

Proof

The trace of a matrix is the sum of its diagonal elements, but transposition leaves the diagonal elements unchanged.

Trace of a product

The next proposition concerns the trace of a product of matrices.

Proposition Let A be a $K	imes L$ matrix and $B$ an $L	imes K$ matrix. Then,[eq14]

Proof

Note that $AB$ is a $K	imes K$ matrix and $BA$ is an $L	imes L$ matrix. Then,[eq15]where in steps $rame{A}$ and $rame{B}$ we have used the definition of matrix product, in particular, the facts that [eq16] is equal to the inner product between the k-th row of A and the k-th column of $B$, and [eq17] is equal to the inner product between the $l$-th row of $B$ and the $l$-th column of A.

Trace of a scalar

A trivial, but often useful property is that a scalar is equal to its trace because a scalar can be thought of as a $1	imes 1$ matrix, having a unique diagonal element, which in turn is equal to the trace.

This property is often used to write inner product as traces.

Example Let A be a $1	imes K$ row vector and $B$ a Kx1 column vector. Then, the product $AB$ is a scalar, and[eq18]where in the last step we have use the previous proposition on the product of traces. Thus, we have been able to write the scalar $AB$ as the trace of the $K	imes K$ matrix $BA$.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let A be a $3	imes 3$ matrix defined by[eq19]Find its trace.

Solution

By summing the diagonal elements, we obtain[eq20]

Exercise 2

Let A be a $K	imes K$ matrix and x a Kx1 vector. Write the product[eq21]as the trace of a product of two $K	imes K$ matrices.

Solution

Since $x^{	op }Ax$ is a scalar, we have that [eq22]Furthermore, $x^{	op }A$ is $1	imes K$ and x is Kx1. Therefore,[eq23]where both $xx^{	op }$ and A are $K	imes K$.

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