29.1.1. Eigenvalues and Eigenvectors

Let A be an n×n square matrix. A scalar λ is called an eigenvalue of A if there exists a non-zero vector x∈Rn such that:

Ax=λx

Here: • λ∈R (or C) is an eigenvalue, • x is a corresponding eigenvector.

This chunk introduces the concept of eigenvalues and eigenvectors. An eigenvalue (λ) is a special kind of scalar associated with a matrix (A). When applied to a vector (x), the matrix transforms this vector into another vector that is simply a scaled version of the original vector by λ. This relationship is expressed by the equation Ax = λx, which shows that the action of matrix A on the vector x yields a vector that retains the same direction but may change in magnitude, depending on the eigenvalue.

Imagine you are pushing a toy car in a straight line. If you push it slightly harder, the car moves faster in the same direction (analogous to 'stretching'). If you push it less hard, it moves slower (analogous to 'compressing'). In this analogy, the direction in which the car moves is like the eigenvector, while how much faster or slower the car moves represents the eigenvalue.

This equation can be rearranged as:

(A−λI)x=0

This is a homogeneous system of linear equations. For a non-trivial solution (x̸=0), the coefficient matrix must be singular, i.e., det(A−λI)=0.

To find the eigenvalues, we rearrange the initial equation into a form that sets up a system of linear equations: (A−λI)x = 0. This system is termed 'homogeneous' because it equals zero. For there to be a solution where x is not zero (non-trivial), the determinant of the matrix (A−λI) must equal zero, indicating that this matrix is 'singular' and lacks an inverse, which leads us to the characteristic polynomial of the matrix.

Consider a tightrope walker trying to balance on a thin wire. If the wire is perfectly straight (analogous to the matrix being non-singular), the walker cannot find a point to position himself where he is balanced (non-zero solution). However, if the wire dips or is off-balance (the matrix becomes singular), there are multiple points where the walker can balance himself without falling, indicating multiple solutions exist.

This is called the characteristic equation, and the polynomial det(A−λI) is the characteristic polynomial.

The equation det(A−λI) = 0 gives us a polynomial known as the characteristic polynomial. By solving this polynomial equation for λ, we find the eigenvalues of the matrix A. The roots of this polynomial correspond to the eigenvalues. Each eigenvalue thus gives us crucial information about how the matrix A operates on space, particularly how it stretches or compresses it in different directions.

Imagine you have a rubber sheet that can be stretched or compressed when pulling on its corners. Each point on the surface represents a vector, and the way the sheet behaves under pressure relates to the eigenvalues. When the sheet is pulled in certain ways (the characteristic polynomial), we can understand how far and in what direction different areas of the sheet will stretch or compress.

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