Trace of a Matrix

The trace of a matrix refers to the sum of the diagonal elements in a square matrix. It is a key concept in linear algebra and is widely used in mathematics, physics, machine learning, and other applied fields.

For a square matrix A of order n×n, the trace is denoted as tr(A) or trace(A) and is defined as the sum of the principal diagonal elements:

tr(A) = a₁₁ + a₂₂ + a₃₃ + ⋯ + a_nn or trace(A) = ⅀ⁿ_n=1 A_nn

where a₁₁, a₂₂, a₃₃, …,​ are the diagonal elements of the matrix A

For example, let us consider a square matrix of order "3 × 3," as shown in the figure given below:
a₁₁, a₁₂, a₁₃,..., a₃₂, and a₃₃ are the entries of the given matrix A.

Now, the trace of matrix "A" is equal to the sum of its principal diagonal elements, i.e., a₁₁, a₂₂, and a₃₃.

Trace of a Matrix Properties

The following are some important properties of a trace of a matrix. Let us consider two square matrices A and B of the same order.

Linearity of the Trace

The sum of the traces of the matrix A and the matrix B is equal to the trace of the matrix that is obtained by the sum of the matrices A and B.

tr(A) + tr(B) = tr (A + B) 

Trace of a Transpose

The trace of a given matrix and its transpose are the same.

tr(A) = tr (A^T)

Trace of a Scalar Multiple

If A is any square matrix of order "n × n" and k is a scalar, then

tr(kA) = k Tr(A)

Trace of a Product

If A is a matrix of order "m × n" and B is a matrix of order "n × m," then the trace of AB is equal to the trace of BA.

tr (AB) = tr (BA)

The above statement is true if both AB and BA are defined.

Trace of an Identity Matrix

 The trace of an identity matrix of order "n × n" is n.

tr(I_n) = n

Trace of a Zero Matrix

The trace of a zero or null matrix of any order is zero.

tr(O) = 0

Articles related to Trace of a Matrix:

• Minors and Cofactors of Determinants
• Adjoint of a Matrix
• Determinant of a Matrix

Solved Examples of Trace of a Matrix

Example 1: Prove that the trace of an identity matrix of order "3 × 3" is 3.

Solution:

Let us consider an identity matrix of order "3 × 3" to prove the trace of an identity matrix of order "3 × 3" is 3.

I₃ =  \left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right]

We know that,
tr(A) = a11 + a22 + a33
tr(A) = 1 + 1 + 1 =3
Hence, proved.

Example 2: Calculate the trace of the matrix given below.

B = \left[\begin{array}{cccc} 1 & -3 & 4 & 7\\ 2 & 11 & -9 & 6\\ 17 & 8 & -5 & 3\\ 5 & -22 & 14 & -4 \end{array}\right]

Solution:

From the given matrix,
b₁₁ = 1, b₂₂ = 11, b₃₃ = −5, and b₄₄ = −4.

We know that,
tr(A) = b₁₁ + b₂₂ + b₃₃ + b₄₄
= 1 + 11 + (−5) + (−4)
= 12 −5 −4 = 12 − 9 = 3

Thus, the trace of the given matrix B is 3.

Example 3: Calculate the trace of the matrix given below.

Solution:

From the given matrix,
a₁₁ = 0, a₂₂ = 24, a₃₃ = 7, a₄₄ = −5, and a₅₅ = 16.

We know that,
tr(A) = a₁₁ + a₂₂ + a₃₃ + a₄₄ + a₅₅
= 0 + 24 + 7 + (−5) + 16
= 47 −5 = 42

Thus, the trace of the given matrix A is 42.

Example 4: If R = P + Q, then prove that tr(R) = tr(P) + tr(Q), where "P, Q, and R" are square matrices of order "2 × 2"

Solution:

Let P = \left[\begin{array}{cc} p_{11} & p_{12}\\ p_{21} & p_{22} \end{array}\right] Q =  \left[\begin{array}{cc} q_{11} & q_{12}\\ q_{21} & q_{22} \end{array}\right]

R = P + Q = \left[\begin{array}{cc} p_{11}+q_{11} & p_{12}+q_{12}\\ p_{21}+q_{21} & p_{22}+q_{22} \end{array}\right]

Now, tr(R) = p₁₁ + q₁₁ + p₂₂ + q₂₂
tr(R) = p₁₁ + p₂₂ + q₁₁ + q₂₂
tr(P) = p₁₁ + p₂₂
tr(Q) = q₁₁ + q₂₂
tr(P) + tr(Q) = p₁₁ + p₂₂ + q₁₁ + q₂₂
tr(P) + tr(Q) = tr(R)

Hence, proved.

Practice Problems on Trace of a Matrix

Question 1: Given the matrixA = \begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix}, calculate the trace of matrix A.

Question 2: Find the trace of the matrix B = \begin{pmatrix} 4 & 1 & 7 \\ 0 & 6 & 8 \\ 2 & 3 & 5 \end{pmatrix}

Question 3: Given the diagonal matrix C = \begin{pmatrix} 9 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{pmatrix} compute the trace of matrix C.

Question 4: If the trace of matrix A = = \begin{pmatrix} 2+x & 3 & 4 \\ 1 & -1 & 2 \\ -5 & 1 & x \end{pmatrix} is 5, then find the value of x.

Answer key

Answer 1: The trace of the matrix equals 10.
Answer 2: The trace of the matrix equals 15.
Answer 3: The trace of the matrix equals 17.
Answer 4: The value of x is 2