characteristic matrix of diagonal element cross-section

Denote by Mn⁒(𝒦) the set of all nΓ—n matrices over 𝒦. Let di:Mn⁒(𝒦)βŸΆπ’¦ be the function which extracts the ith diagonal element of a matrix. Finally denote by [n] the set {1,…,n}.


Let K be a field. Let a sequence A1,…,Ar∈Mn⁒(K) of upper triangular matricesMathworldPlanetmath be given, and denote by AβŠ†Mn⁒(K) the unital algebra generated by these matrices. For every sequence Ξ»1,…,Ξ»r∈K of scalars there exists a matrix C∈A such that

di⁒(C)={1ifΒ di⁒(Ak)=Ξ»kΒ for allΒ k∈[r],0π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’

for all i∈[n].

A diagonal element of an upper triangular matrix is of course an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of that matrix, so the particular Ξ»1,…,Ξ»r that one plugs into this lemma is typically either a sequence of eigenvalues for the given matrices, or a sequence of values that one thinks may be eigenvalues for these matrices. The β€œcross-section” in the title refers to that there is one Ξ»k for each matrix Ak.

The result gets more interesting if one knows something more about π’œ than what was explicitly required above. A particular example is that if the given matrices commute then π’œ will be a commutative ( algebra and consequently C will commute with all of the given matrices as well.


Let S be the set of those row indices i for which di⁒(C) should be 0, i.e.,

S={i∈[n]|di⁒(Ak)β‰ Ξ»kΒ for someΒ k∈[r]}⁒,

and for each i∈S let ki∈[r] be such that di⁒(Aki)β‰ Ξ»ki. Define

Bi={Aki-di⁒(Aki)⁒Iif i∈S,Iotherwise

for all i∈[n] and let B=∏i=1nBi. Since all the matrices involved are upper triangular, a diagonal element in B is simply the product of the corresponding diagonal elements in all of B1,…,Bn. If i∈S then di⁒(Bi)=di⁒(Aki-di⁒(Aki)⁒I)=di⁒(Aki)-di⁒(Aki)=0 and thus di⁒(B)=0 for i∈S. If instead i∈[n]βˆ–S then


This b is by definition of kj nonzero, and since it is independent of i it follows that it is the only nonzero value of a diagonal element of B. Hence the wanted C=b-1⁒B. ∎

Title characteristic matrix of diagonal element cross-section
Canonical name CharacteristicMatrixOfDiagonalElementCrosssection
Date of creation 2013-03-22 15:29:45
Last modified on 2013-03-22 15:29:45
Owner lars_h (9802)
Last modified by lars_h (9802)
Numerical id 4
Author lars_h (9802)
Entry type Theorem
Classification msc 15A30