Peilimatriisit

Peilimatriisit, known in English as mirror matrices or reflection matrices, are a type of square matrix used in linear algebra and computer graphics to represent a reflection across a line or plane. These matrices are characterized by having eigenvalues of 1 and -1. Specifically, for a reflection in n-dimensional space, the matrix will have n-1 eigenvalues equal to 1 and one eigenvalue equal to -1. When a vector is multiplied by a peilimatriisi, it is transformed into its reflection across the designated line or plane.

The geometric interpretation of a peilimatriisi is crucial to understanding its function. A reflection preserves distances and angles, meaning it is an orthogonal transformation. The determinant of a peilimatriisi is always -1, which signifies that the orientation of the space is reversed by the reflection. For example, in two dimensions, a peilimatriisi can represent a reflection across the x-axis, the y-axis, or any other line passing through the origin. In three dimensions, it can represent a reflection across a plane passing through the origin. The construction of a peilimatriisi often involves the use of eigenvectors and eigenvalues, where the eigenvector corresponding to -1 defines the direction of reflection.