page_title: Eigenvalues and eigenvectors
hierarchical_section_title: Eigenvalues and eigenvectors
caption_attribution_description: On the left, a reproduction of the Mona Lisa with a representation of two vectors superimposed on top of it. The red arrow represents a vector in three dimensions pointing directly up on the plane of the painting and the blue arrow represents a unique arbitrary vector on the painting. On the right is the result of a shear transformation on the image on the left. The blue arrow now represents a different vector, while the red one still represents the same vector. For this reason, the red vector is an eigenvector of the transformation, and the blue vector is not.
context_page_description: Linear algebra talks about types of functions called transformations. In that context, an eigenvector is a vector—different from the null vector—which does not change direction after the transformation. The vector may change its length, or become zero. The eigenvalue is the value of the vector's change in length, and is typically denoted by the symbol. The word "eigen" is a German word, which means "own" or "typical".
context_section_description: Linear algebra talks about types of functions called transformations. In that context, an eigenvector is a vector—different from the null vector—which does not change direction after the transformation (except if the transformation turns the vector to the opposite direction). The vector may change its length, or become zero ("null"). The eigenvalue is the value of the vector's change in length, and is typically denoted by the symbol . The word "eigen" is a German word, which means "own" or "typical".
caption: Illustration of a transformation (of Mona Lisa): The image is changed in such a way that the red arrow (vector) does not change its direction, but the blue one does. The red vector therefore is an eigenvector of this transformation, the blue one is not. Since the red vector does not change its length, its eigenvalue is 1. The transformation used is called shear mapping.
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