Eigenértékei

Eigenértékei, or eigenvalues, are fundamental concepts in linear algebra. They represent scalars that describe how a particular linear transformation stretches or compresses an eigenvector. When a linear transformation is applied to a vector, the result is a scaled version of that same vector if the vector is an eigenvector of the transformation. The eigenvalue is the scaling factor. Mathematically, for a square matrix A and a non-zero vector v, if Av = λv, then λ is an eigenvalue and v is the corresponding eigenvector.

Eigenvalues and eigenvectors are crucial for understanding the behavior of linear systems. They are used in a wide variety of applications, including principal component analysis (PCA) in data science, quantum mechanics in physics, and stability analysis in engineering. Finding eigenvalues typically involves solving the characteristic equation, which is derived from the determinant of the matrix (A - λI), where I is the identity matrix. The roots of this polynomial equation are the eigenvalues. Once the eigenvalues are known, the corresponding eigenvectors can be found by solving the system of linear equations (A - λI)v = 0.