common eigenvector of a diagonal element cross-section
Denote by Mn(𝒦) the set of all n×n matrices over 𝒦. Let di:Mn(𝒦)⟶𝒦 be the function which extracts the ith diagonal element of a matrix, and let εi:𝒦n⟶𝒦 be the function which extracts the ith of a vector. Finally denote by [n] the set {1,…,n}.
Theorem 1.
Let K be a field.
For any sequence A1,…,Ar∈Mn(K) of upper
triangular pairwise commuting matrices and every row index
i∈[n], there exists u∈Kn∖{0}
such that
Ak𝐮=di(Ak)𝐮 | (1) |
Proof.
Let for all , so that the
problem is to find a common eigenvector of
whose corresponding eigenvalue
for is . It is
sufficient to find such a common eigenvector in the case that
is the least for which
for all , because if some smaller also
has this property then one can solve the corresponding problem for
the submatrices
consisting of rows and columns
through of , and then pad the common eigenvector
of these submatrices with zeros to get a common eigenvector of the
original .
By the existence of a
characteristic matrix of a diagonal element
cross-section (http://planetmath.org/CharacteristicMatrixOfDiagonalElementCrossSection)
there exists a matrix in the unital algebra
generated by such that if
for all , and
otherwise; in other words that matrix satisfies and
for all . Since it is also upper triangular it
follows that the matrix has
rank (http://planetmath.org/RankLinearMapping)
, so the kernel of this
matrix is one-dimensional. Let be such that ; it is easy to see that
this is always possible (indeed, the only vector in this nullspace
with th
is the zero vector
). This is the
wanted eigenvector.
To see that it is an eigenvector of , one may first observe
that commutes with this , since the unital algebra of
matrices to which belongs is
commutative (http://planetmath.org/Commutative). This implies that
since . As is
one-dimensional it follows that
for some . Since is upper triangular
and this must furthermore satisfy
, which is
indeed what the eigenvalue was claimed to be.
∎
Title | common eigenvector of a diagonal element cross-section |
---|---|
Canonical name | CommonEigenvectorOfADiagonalElementCrosssection |
Date of creation | 2013-03-22 15:30:41 |
Last modified on | 2013-03-22 15:30:41 |
Owner | lars_h (9802) |
Last modified by | lars_h (9802) |
Numerical id | 4 |
Author | lars_h (9802) |
Entry type | Theorem |
Classification | msc 15A18 |
Related topic | CommutingMatrices |