Editing Linear algebra (section)

===Eigenvalues and eigenvectors===
{{main|Eigenvalues and eigenvectors}}
If {{mvar|f}} is a linear endomorphism of a vector space {{mvar|V}} over a field {{mvar|F}}, an ''eigenvector'' of {{mvar|f}} is a nonzero vector {{mvar|v}} of {{mvar|V}} such that {{math|1=''f''(''v'') = ''av''}} for some scalar {{mvar|a}} in {{mvar|F}}. This scalar {{mvar|a}} is an ''eigenvalue'' of {{mvar|f}}.

If the dimension of {{mvar|V}} is finite, and a basis has been chosen, {{mvar|f}} and {{mvar|v}} may be represented, respectively, by a square matrix {{mvar|M}} and a column matrix {{mvar|z}}; the equation defining eigenvectors and eigenvalues becomes
:<math>Mz=az.</math>
Using the [[identity matrix]] {{mvar|I}}, whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten
:<math>(M-aI)z=0.</math>
As {{mvar|z}} is supposed to be nonzero, this means that {{math|''M'' – ''aI''}} is a [[singular matrix]], and thus that its determinant {{math|det (''M'' − ''aI'')}} equals zero. The eigenvalues are thus the [[root of a function|roots]] of the [[polynomial]]
:<math>\det(xI-M).</math>
If {{mvar|V}} is of dimension {{mvar|n}}, this is a [[monic polynomial]] of degree {{mvar|n}}, called the [[characteristic polynomial]] of the matrix (or of the endomorphism), and there are, at most, {{mvar|n}} eigenvalues.

If a basis exists that consists only of eigenvectors, the matrix of {{mvar|f}} on this basis has a very simple structure: it is a [[diagonal matrix]] such that the entries on the [[main diagonal]] are eigenvalues, and the other entries are zero. In this case, the endomorphism and the matrix are said to be [[diagonalizable matrix|diagonalizable]]. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after [[field extension|extending]] the field of scalars. In this extended sense, if the characteristic polynomial is [[square-free polynomial|square-free]], then the matrix is diagonalizable.

A [[symmetric matrix]] is always diagonalizable. There are non-diagonalizable matrices, the simplest being
:<math>\begin{bmatrix}0&1\\0&0\end{bmatrix}</math>
(it cannot be diagonalizable since its square is the [[zero matrix]], and the square of a nonzero diagonal matrix is never zero).

When an endomorphism is not diagonalizable, there are bases on which it has a simple form, although not as simple as the diagonal form. The [[Frobenius normal form]] does not need to extend the field of scalars and makes the characteristic polynomial immediately readable on the matrix. The [[Jordan normal form]] requires to extension of the field of scalar for containing all eigenvalues and differs from the diagonal form only by some entries that are just above the main diagonal and are equal to 1.