Eigenvalues and eigenvectors: Difference between revisions

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[[Image:Mona Lisa with eigenvector.png|thumb|270px|Fig. 1. In this [[shear (mathematics)|shear]] transformation of the [[Mona Lisa]], the picture was deformed in such a way that its central vertical axis (red vector) was not modified, but the diagonal vector (blue) has changed direction. Hence the red vector is an <font color="#CC1111">'''eigenvector'''</font> of the transformation and the blue vector is <font color="darkblue">not</font>. Since the red vector was neither stretched nor compressed, its '''eigenvalue''' is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue. Together with the zero-vector, they form the '''eigenspace''' for this eigenvalue.]]

In [[mathematics]], a [[Vector (spatial)|vector]] may be thought of as an arrow. It has a length, called its ''magnitude'', and it points in some particular ''direction''. A [[linear transformation]] may be considered to operate on a vector to change it, usually changing both its magnitude and its direction. An {{Audio|De-eigenvector.ogg|'''eigenvector'''}} of a given linear transformation is a vector which is simply multiplied by a constant called the {{Audio-nohelp|De-eigenvalue.ogg|'''eigenvalue'''}} during that transformation. The direction of the eigenvector is either unchanged by that transformation (for positive eigenvalues) or reversed (for negative eigenvalues).

For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector stays the same, while an eigenvalue of &minus;1 means that the eigenvector is reversed in direction. An '''eigenspace''' of a given transformation is the set of all eigenvectors of that transformation that have the same eigenvalue, together with the zero vector (which has no direction). An '''eigenspace''' is an example of a [[linear subspace|subspace]] of a [[vector space]].