Editing Jordan normal form (section)

=== Invariant subspace decompositions ===

The Jordan form of a ''n'' × ''n'' matrix ''A'' is block diagonal, and therefore gives a decomposition of the ''n'' dimensional Euclidean space into invariant subspaces of ''A''. Every Jordan block ''J''<sub>''i''</sub> corresponds to an invariant subspace ''X''<sub>''i''</sub>. Symbolically, we put

:<math>\mathbb{C}^n = \bigoplus_{i = 1}^k X_i</math>

where each ''X''<sub>''i''</sub> is the span of the corresponding Jordan chain, and ''k'' is the number of Jordan chains.

One can also obtain a slightly different decomposition via the Jordan form. Given an eigenvalue ''λ''<sub>''i''</sub>, the size of its largest corresponding Jordan block ''s''<sub>''i''</sub> is called the '''index''' of ''λ''<sub>''i''</sub> and denoted by {{math|''v''(''λ''<sub>''i''</sub>)}}. (Therefore, the degree of the minimal polynomial is the sum of all indices.) Define a subspace ''Y''<sub>''i''</sub> by

:<math> Y_i = \ker(\lambda_i I - A)^{v(\lambda_i)}.</math>

This gives the decomposition

:<math>\mathbb{C}^n = \bigoplus_{i = 1}^l Y_i</math>

where {{mvar|l}} is the number of distinct eigenvalues of ''A''. Intuitively, we glob together the Jordan block invariant subspaces corresponding to the same eigenvalue. In the extreme case where ''A'' is a multiple of the identity matrix we have ''k'' = ''n'' and ''l'' = 1.

The projection onto ''Y<sub>i</sub>'' and along all the other ''Y<sub>j</sub>'' ( ''j'' ≠ ''i'' ) is called '''the spectral projection of ''A'' at {{math|v<sub>''i''</sub>}}''' and is usually denoted by '''''P''(''λ''<sub>''i''</sub> ; ''A'')'''. Spectral projections are mutually orthogonal in the sense that {{math|1=''P''(''λ''<sub>''i''</sub> ; ''A'') ''P''(v<sub>''j''</sub> ; ''A'') = 0}} if ''i'' ≠ ''j''. Also they commute with ''A'' and their sum is the identity matrix. Replacing every v<sub>''i''</sub> in the Jordan matrix ''J'' by one and zeroing all other entries gives {{math|''P''(v<sub>''i''</sub> ; ''J'')}}, moreover if ''U J U''<sup>−1</sup> is the similarity transformation such that ''A'' = ''U J U''<sup>−1</sup> then ''P''(''λ''<sub>''i''</sub> ; ''A'') = ''U P''(''λ''<sub>''i''</sub> ; ''J'') ''U''<sup>−1</sup>. They are not confined to finite dimensions. See below for their application to compact operators, and in [[holomorphic functional calculus]] for a more general discussion.

Comparing the two decompositions, notice that, in general, {{math|''l'' ≤ ''k''}}. When ''A'' is normal, the subspaces ''X''<sub>''i''</sub>'s in the first decomposition are one-dimensional and mutually orthogonal. This is the [[spectral theorem]] for normal operators. The second decomposition generalizes more easily for general compact operators on Banach spaces.

It might be of interest here to note some properties of the index, {{math|''ν''(''λ'')}}. More generally, for a complex number ''λ'', its index can be defined as the least non-negative integer {{math|''ν''(''λ'')}} such that

:<math>\ker(A-\lambda I)^{\nu(\lambda)} = \ker(A-\lambda I)^m, \; \forall m \geq \nu(\lambda) .</math>

So {{math|''ν''(v) > 0}} if and only if ''λ'' is an eigenvalue of ''A''. In the finite-dimensional case, {{math|''ν''(v) ≤}} the algebraic multiplicity of {{math|v}}.