Trace of a Matrix

What is a square matrix?

A square matrix is defined as a matrix that has an equal number of rows and columns. For example, if the order of a square matrix is “3 × 3,” then it has three rows and three columns.

What is meant by the trace of a matrix?

The trace of a matrix is defined as the sum of the principal diagonal elements of a square matrix. It is usually represented as tr(A), where A is any square matrix of order “n × n.”

What is the trace of a null matrix?

The trace of a zero or null matrix of any order is zero, i.e., tr(O) = 0.

What is the trace of an identity matrix of order n?

The trace of an identity matrix of order “n × n” is n, i.e., tr (I_n) = n.

tr (I_n) = 1 + 1+ 1 + …+ 1 (n times) = n

What is the definition of Trace of a Matrix?

Trace of a matrix is defined as the sum of the principal diagonal elements of a square matrix. It is usually represented as tr(A), where A is any square matrix of order “n × n.”

What is the trace of a matrix?

The trace of a matrix, specifically a square matrix, is the sum of its diagonal elements. This property holds significance in various mathematical computations and applications​.

How do you calculate the trace of a matrix?

To calculate the trace of a matrix, you simply sum up all the diagonal entries from the upper-left corner to the bottom-right corner of the matrix​.

What properties does the trace of a matrix have?

The trace of a matrix has several important properties:

• It is invariant under cyclic permutations of matrix products.
• The trace of a matrix and its transpose are the same.
• It behaves linearly, meaning the trace of a sum of matrices equals the sum of their traces, and the trace of a scalar multiplied by a matrix equals the scalar times the trace of the matrix​.

Is the trace of a matrix affected by a change in basis?

No, the trace of a matrix is independent of the basis. This means it remains the same regardless of how the matrix is represented, as it is intrinsically linked to the eigenvalues of the matrix, which are basis-independent​.

Can the trace of a matrix be negative?

Yes, the trace of a matrix can be negative if the matrix has negative diagonal elements. The trace is simply the sum of these diagonal values, which can be positive, negative, or zero​.

How is the trace related to the eigenvalues of a matrix?

The trace of a matrix equals the sum of its eigenvalues. This relationship is often used to derive properties of matrices in various fields of study​.

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 “m by n” matrix when it has “m” rows and “n” columns and is written as an “m × n” matrix. For example, if a matrix has three rows and four columns, then the order of the matrix is “3 × 4.” 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.