What do eigenvalues of a matrix represent?

Eigenvalues of a matrix are fundamental in understanding how linear transformations affect vectors in a specific space. They represent the magnitude by which a given vector is stretched or compressed during the transformation defined by the matrix. More precisely, when a matrix acts on its eigenvector, the output is a scalar multiple of that eigenvector, where the scalar is the corresponding eigenvalue.

This means that an eigenvalue provides insight into the scaling effect of the transformation in the direction of its eigenvector. If the eigenvalue is greater than one, it indicates stretching; if it is between zero and one, it indicates compressing; and if the eigenvalue is negative, it also indicates a reflection along with stretching or compressing.

Understanding the significance of the eigenvalues aids in various applications, including stability analysis, transformations in graphics, and solving systems of differential equations. This clear relationship between eigenvalues and the geometric interpretation of the transformation is why option B effectively captures the essence of what eigenvalues represent in linear algebra.