Eigenvalue Equation

In linear algebra, the eigenvalue equation plays a fundamental role in the study of linear transformations and matrices. Given a square matrix $A$, an eigenvalue $\lambda$ for $A$ is a scalar such that there exists a non-zero vector $\mathbf{v}$ satisfying the equation:

$A\mathbf{v}=\lambda \mathbf{v}$

Here, $\mathbf{v}$ is called an eigenvector corresponding to the eigenvalue $\lambda$. This relationship can be rewritten as:

$(A-\lambda I)\mathbf{v}=0$

where $I$ is the identity matrix of the same size as $A$.

Eigenvalues and eigenvectors have numerous applications in mathematics and various scientific fields. They are used in solving systems of linear differential equations, analyzing structures in engineering, and studying quantum mechanics.

Properties of Eigenvalue Equation:

• Eigenvalues can be complex numbers.
• The sum of all eigenvalues of a matrix is equal to the trace of the matrix.
• The product of all eigenvalues of a matrix is equal to the determinant of the matrix.

Example:

Consider the matrix

$A=\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$

To find the eigenvalues of $A$, we need to solve the characteristic equation:

$\text{det}(A-\lambda I)=0$

Substitute $A$ into the equation, we get:

$\text{det}\left(\begin{array}{cc}3-\lambda & 1\\ 1& 3-\lambda \end{array}\right)=0$

Solving this equation gives us the eigenvalues $\lambda =2,4$.

Historical Context:

The concept of eigenvalues and eigenvectors was introduced by the German mathematician David Hilbert in the early 20th century. Eigenvalues have since become a central tool in many areas of mathematics, physics, and engineering.

Real-Life Example:

In physics, when studying the vibrational modes of a system, eigenvalues and eigenvectors can help determine the frequencies at which the system oscillates naturally. This information is crucial in understanding the stability and behavior of the system under different conditions.

---

Exam Questions:

1. Given the matrix $B=\left(\begin{array}{cc}1& 0\\ 0& -2\end{array}\right)$, find the eigenvalues and corresponding eigenvectors.

2. Prove that the sum of the eigenvalues of a $3\times 3$ matrix is equal to the sum of its diagonal elements.

3. Discuss a real-life application where eigenvalues and eigenvectors are used in solving practical problems.