Drazin Inverse of Jordan Form Matrix

Teduh Wulandari Mas’oed1,a) and Agah D. Garnadi1,b)

1Department of Mathematics, FMIPA-IPB, Jl Meranti Kampus Darmaga, Institut Pertanian Bogor, Indonesia 16680.

a)Corresponding author: [email protected] b)[email protected]

Abstract. One type of Generalized Inverse is Drazin Inverse, or some people say the spectral inverse. In this note, we pointed out that through Jordan Form Matrices, the Drazin Inverse is well understood. The authors would like dedicated this note to Prof. Irawati and Prof. Sri Wahyuni on their 60th Birthday. Keywords: Generalized Inverse, Drazin Inverse, Jordan Form, Square Matrices

INTRODUCTION

In [1] Drazin was the first to mention a type of pseudo-inverse in context of assosiative rings and semigroups. Because the set of square matrices with respect to matrix multiplication is a semigroup and together with matrix addition is an associative rings, matrix theorist began examine of pseudo-inverse, which they coined this type of pseudo-inverse as Drazin-inverse. Ben-Israel and Greville at [2] devoted a chapter considering Drazin-inverse. Campbell and Meyer [3] dedicated several chapters about it, mostly motivated by applications. Ben-Israel and Greville [2] treated Drazin- inverse from the matrix/linear algebra stand point. They handle several algorithm to compute Drazin-inverse. It was J.H. Wilkinson [4], who first pointed out how to compute Drazin-inverse from practical computation point of view. This note is an exposition how to compute Drazin-inverse algorithmically building from basic linear algebra. To achieve this, the exposition will be ordered as follows : First we will stated the object of interest which is the Drazin inverse. In the second part we demonstrate that for matrix of Jordan form, its Drazin inverse is established for its existence and uniqueness. Lastly, we provides a few examples to illustrate the idea.

DRAZIN INVERSE

In this section we will discuss the definition of Drazin Inverse. Let An×n be a square matrix with real entries. Based on [2] the following definitions can be obtained Definition Xn×n is called a Drazin Inverse of An×n if it satisfies

AkXA = Ak, for some k ∈ {1, 2, 3, ··· , n} (1) XAX = X (2) AX = XA (3) JORDAN FORM MATRIX AND ITS DRAZIN INVERSE

JORDAN FORM MATRIX To begin with, let us define what we mean by a Jordan Block. In his book, Pullman [5] defines Jordan Block as follows. Definition A Jordan block Jn(λn) is n × n matrix

1. each diagonal entry is λn 2. each super diagonal entry 1 3. all other entries zero

This definition of Jordan Block was then developed by [5] into a definition for the Jordan Matrix. Definition[5] Jn×n is Jordan Matrix iff Jn×n is direct sum of Jordan blocks

Mm Jn×n = Jni×ni (λi), i=1

where Pm n = n i=1 i L Let An×n = (ai, j) and Bn×n = (bi, j). A direct sum of An×n Bm×m is C(m+n)×(m+n) for which

• ci j = ai j for all 1 ≤ i, j ≤ n

• ci j = b(i−n)( j−n) for all n + 1 ≤ i, j ≤ m + n

• ci j = 0 for all other i, j L L L L Let Ani×ni , Bn2×n2 , and Cn3×n3 , by defining A B C = (A B) C We can extend the summand inductively to obtain

Mn M M M Ai = A1 A2 ··· An i=1

Wilkinson explained in his book [4] that each square matrix has a unique Jordan Form Matrix up to permutation.

DRAZIN INVERSE OF Jn×n This segment will talk about how to get Darzin Inverse from Jordan Matrix. Let : Mm Jn×n = Jni×ni (λi), i=1 Pm where i=1 ni = n Define : ( −1 D Jni×ni (λi) , if λi , 0, Jni×ni (λi) = (0)ni×ni , else Then: m D M D Jn×n = Jni×ni (λi) , i=1 Pm where i=1 ni = n EXAMPLES

In this section an example is given of forming Drazin inverses from a Jordan Matrix. " # " # 1 1 0 1 J = ; J = 1 0 1 2 0 0 " # " # 1 −1 0 0 JD = ; JD = 1 0 1 2 0 0  1 1 0 0    M  0 1 0 0  A = J J =   1 2  0 0 0 1     0 0 0 0 

 1 −1 0 0    M  0 1 0 0  AD = JD JD =   , k = 1. 1 2  0 0 0 0     0 0 0 0 

CONCLUSION

We have shown that for every matrix of Jordan form, it’s Drazin inverse is easily computed. Hence, since every square matrix have a unique Jordan form up to permutation [6], then every matrix have a unique Drazin Inverse.

ACKNOWLEDGEMENT

This note is a response to queries from Profs. Pudji Astuti Walujo and Indah Widjajanti, of our works on Leslie Matrices during KNM 2018 in Malang.

REFERENCES

[1] M. Drazin, The American Mathematical Monthly 65, 506–514 (1958). [2] A. Ben-Israel and T. Greville, Generalized inverses: theory and applications, Vol. 15 (Springer Science & Business Media, 2003). [3] S. Campbell and C. Meyer Jr, Recent applications of the Drazin inverse, Recent Applications of Generalized Inverses, Ed (1982). [4] J. H. Wilkinson, “Note on the practical significance of the drazin inverse,” in Recent Applications of General- ized Inverses, Res. Notes in Math., Vol. 66, edited by S. L. Campbell (Pitman Publishing,London, UK, 1982) p. 8299. [5] N. J. Pullman, Matrix theory and its applications (M. Dekker,, 1976). [6] I.Gohberg and S. Goldberg, The Am. Math. Monthly 103(2), 157–159 (1996).