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.
- 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)
- The trace of a given matrix and its transpose are the same.
tr(A) = tr (AT)
- If A is any square matrix of order βn Γ nβ and k is a scalar, then
tr(kA) = k Tr(A)
- If A is a matrix of order βm Γ nβ and B is a matrix of order βn Γ m,β 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.
- The trace of an identity matrix of order βn Γ nβ is n.
tr(In) = n
- The trace of a zero or null matrix of any order is zero.
tr(O) = 0
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Trace of a Matrix
Trace of a Matrix: A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an βm by nβ matrix when it has βmβ rows and βnβ columns and is written as an βm Γ nβ matrix. For example, if a matrix has three rows and four columns, then the order of the matrix is β3 Γ 4.β We have different types of matrices, such as rectangular, square, triangular, symmetric, etc.
In this article, we will learn about the Trace of a matrix, along with its definition, Trace of a Matrix properties, and Trace of a Matrix examples.