Question

Prove that the trace of a matrix equals the sum of all \(n\) of its eigenvalues and the determinant is their product.

Short answer

Question: Prove that the trace of a square matrix equals the sum of its eigenvalues, and the determinant equals the product of its eigenvalues. Answer: We can prove this by following these steps: 1. Define the trace and determinant of a matrix. 2. Familiarize ourselves with eigenvalues and eigenvectors. 3. Develop an understanding of the characteristic polynomial and its relation to eigenvalues. 4. Show that the trace of a matrix equals the sum of its eigenvalues. 5. Show that the determinant of a matrix equals the product of its eigenvalues. By considering the characteristic polynomial and its coefficients, we can determine that the trace of a matrix is equal to the sum of its eigenvalues, and the determinant is equal to the product of its eigenvalues. This conclusion is supported by understanding the properties of eigenvalues, eigenvectors, and the characteristic polynomial.

Step-by-step solution

Define the trace and determinant of a matrix

The trace of a square matrix \(A\) is the sum of its diagonal elements denoted as \(tr(A)\): $$tr(A) = \sum_{i=1}^{n} a_{ii}$$ where \(a_{ii}\) is the element in the \(i\)-th row and \(i\)-th column of matrix \(A\). The determinant of a square matrix \(A\) is a scalar number, denoted as \(det(A)\) or \(|A|\). It represents a matrix's scaling factor and can be computed using various methods, such as the cofactor expansion or row reduction.

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are essential in understanding the properties of matrices. Given a square matrix \(A\), if there is a non-zero vector \(v\) and a scalar \(\lambda\), such that the following relation holds: $$A v = \lambda v$$ then \(\lambda\) is called the eigenvalue of matrix \(A\), and \(v\) is called an eigenvector corresponding to the eigenvalue \(\lambda\).

Characteristic polynomial

In order to find the eigenvalues of a matrix, we use the characteristic polynomial \(p_A(\lambda)\). The characteristic polynomial of a square matrix \(A\) is defined as: $$p_A(\lambda)=det(A-\lambda I)$$ where \(I\) is the identity matrix. The roots of the characteristic polynomial are the eigenvalues of the matrix \(A\).

Prove the trace equals the sum of the eigenvalues

Let's consider the characteristic polynomial of the matrix \(A\) and its coefficients. We know that it is an \(n\)-degree polynomial: $$p_A(\lambda) = \lambda^n - c_1 \lambda^{n-1} + c_2 \lambda^{n-2} - \cdots +(-1)^n c_n$$ Notice that the coefficient \(c_1\) is the negation of the trace of the matrix A, i.e., \(c_1 = - tr(A)\). Since \(c_1\) is the sum of all eigenvalues' negations, we can conclude that the sum of all eigenvalues of \(A\) is equal to the trace of \(A\).

Prove the determinant equals the product of the eigenvalues

Let \(\lambda_1, \lambda_2, \cdots, \lambda_n\) be the eigenvalues of matrix \(A\). By the definition of the characteristic polynomial \(p_A(\lambda)\), we have: $$p_A(0)=det(A-0I)=det(A)$$ Now, substitute \(\lambda = 0\) in the expanded form of the characteristic polynomial: $$p_A(0) = 0^n - c_1 0^{n-1} + c_2 0^{n-2} - \cdots +(-1)^n c_n=(-1)^n c_n$$ Hence, \(det(A)=(-1)^n c_n\). The coefficient \(c_n\) is the product of all eigenvalues of the matrix \(A\). This means that the determinant of \(A\) equals the product of its eigenvalues. In conclusion, we have proved that the trace of a matrix equals the sum of all \(n\) eigenvalues and the determinant equals their product.

Key concepts

Trace of a Matrix

Understanding the trace of a matrix is essential for various areas of mathematics and physics. Simply put, the trace is the sum of all the diagonal elements in a square matrix. For matrix \(A\), the trace is denoted as \(tr(A)\) and can be mathematically expressed as:
\[tr(A) = \bigg\bigg\bigg\sum_{i=1}^{n} a_{ii}\bigg\bigg\bigg\]
where \(a_{ii}\) represents the element positioned in the \(i\)-th row and \(i\)-th column. As described in the exercise, the trace of a matrix has a deep connection with eigenvalues, as it is equal to the sum of all eigenvalues of the matrix. This property is not just a numerical coincidence but arises from the fundamental structure of the matrix and has significant implications in matrix algebra.

Determinant of a Matrix

The determinant is another key concept when it comes to square matrices, symbolized as \(det(A)\) or \(|A|\). It gives important information about the matrix, such as whether it is invertible or not. The determinant can be seen as a scaling factor for the transformation described by the matrix. In the context of eigenvalues, the determinant proves to be the product of all eigenvalues of the matrix. To compute the determinant, methods like cofactor expansion or row reduction could be used, both offering a systematic approach to find this scalar quantity that holds profound meaning in system solutions and matrix properties.

Characteristic Polynomial

The characteristic polynomial is pivotal in finding the eigenvalues of a matrix. For a given square matrix \(A\), it is denoted \(p_A(\lambda)\), yielding a polynomial equation, which roots are precisely the eigenvalues of \(A\). Mathematically, it is expressed as:
\[p_A(\lambda)=det(A-\lambda I)\]
In this equation, \(I\) represents the identity matrix, and \(\lambda\) represents any scalar value. When we set \(\lambda\) to zero, we actually compute the determinant of the matrix, directly attributing to the fascinating relationship between the determinant and the eigenvalues, as explored in the exercise solution.

Matrix Algebra

Matrix algebra is a robust branch of mathematics that deals with matrices and the various operations that can be performed on them. It encompasses concepts such as addition, multiplication, transpose, inverse, trace, and determinant, along with eigenvalues and eigenvectors. These operations and properties enable us to solve systems of linear equations, transform geometric objects, and much more. The understanding of how the trace and determinant relate to eigenvalues is a fundamental part of this discipline. These relationships not only provide a deeper insight into the behavior of matrices but also assist in myriad applications in engineering, data analysis, and beyond.