The Jordan forms of AB and BA

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1 Electronic Journal of Linear Algebra Volume 18 Volume 18 (29) Article The Jordan forms of AB and BA Ross A. Lippert Gilbert Strang Follow this and additional works at: Recommended Citation Lippert, Ross A. and Strang, Gilbert. (29), "The Jordan forms of AB and BA", Electronic Journal of Linear Algebra, Volume 18. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June 29 THE JORDAN FORMS OF AB AND BA ROSS A. LIPPERT AND GILBERT STRANG Abstract. The relationship between the Jordan forms of the matrix products AB and BA for some given A and B was first described by Harley Flanders in Their non-zero eigenvalues and non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1. We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic. Key words. Jordan form, Weyr characteristic, eigenvalues AMS subject classifications. 15A21, 15A18 1. Introduction. Suppose A and B are n n complex matrices, and suppose A is invertible. Then AB = A(BA)A 1. The matrices AB and BA are similar. They have the same eigenvalues with the same multiplicities, and more than that, they have the same Jordan form. This conclusion is equally true if B is invertible. IfbothA and B are singular (and square), a limiting argument involving A + ɛi is useful. In this case AB and BA still have the same eigenvalues with the same multiplicities. What the argument does not prove (because it is not true) is that AB is similar to BA. Their Jordan forms may be different, in the sizes of the blocks associated with the eigenvalue λ =. This paper studies that difference in the block sizes. The block sizes can increase or decrease by 1. This is illustrated by an example in which AB has Jordan blocks of sizes 2 and 1 while BA has three 1 by 1 blocks. We could begin with Jordan matrices A and B: 1 1 A = and B = The product AB is zero. The product BA also has a triple zero eigenvalue but the Received by the editors May 5, 29. Accepted for publication May 28, 29. Handling Editor: Roger A. Horn. 123 West 92 Street #1, New York, NY 125, USA (ross.lippert@gmail.com). MIT Department of Mathematics, 77 Massachusetts Avenue, Building Room 2-24, Cambridge, MA 2139, USA (gs@math.mit.edu). 281

3 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June R.A. Lippert and G. Strang rank is 1. In fact, BA is in Jordan form: 1 BA = A different 3 by 3 example illustrates another possibility: 1 1 A = 1 and B = 1 with 1 1 AB = and BA = 1 Those examples show all the possible differences for n =3,whenAB is nilpotent. More generally, we want to find every possible pair of Jordan forms for AB and BA, for any n m matrix A and m n matrix B over an algebraically closed field. The solution to this problem, generalized to matrices over an arbitrary field, was given over 5 years ago by Harley Flanders [3], with subsequent generalizations and specializations [4, 6]. In this article, we give a novel elementary proof by using the Weyr characteristic. 2. The Weyr Characteristic. There are two dual descriptions of the Jordan block sizes for a specific eigenvalue. We can list the block dimensions σ i in decreasing order, giving the row lengths in Figure 2.1. This is the Segre characteristic. We can σ 1 =4 σ 2 =4 σ 3 =2 σ 4 =1 ω 1 =4 ω 2 =3 ω 3 =2 ω 4 =2 Fig A tableau representing the Jordan structure J 4 J 4 J 2 J 1. also list the column lengths ω 1,ω 2,... (they automatically come in decreasing order).

4 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June 29 Jordan forms of AB and BA 283 This is the Weyr characteristic. By convention, we define σ i and ω i for all i>by setting them to for sufficiently large i. Ifweconsider{σ i } and {ω j } to be partitions of their common sum n, then they are conjugate partitions: σ i counts the number of j s for which ω j i and vice versa. The relationship between conjugate partitions {σ i } and {ω i } is compactly summarized by ω σi i>ω σi+1 (or by σ ωi i>σ ωi+1), the first inequality making sense only when σ i >. Tyingthetwodescriptionsto linear algebra is the nullity index ν j : ν j (A) =dimnull(a j ) = dimension of the nullspace of A j (with ν (A) =). Thus ν j counts the number of generalized eigenvectors for λ =withheight j or less. In the example in Figure 2.1, ν,...,ν 5 are, 4, 7, 9, 11. Then ω j = ν j ν j 1 counts the number of Jordan blocks of size i or greater for λ =. Further exposition of the Weyr characteristic can be found in [5] and some geometric applications in [1, 2]. Our main theorem is captured in the statement that ω i (BA) ω i+1 (AB). Reversing A and B gives a parallel inequality that we re-index as ω i 1 (AB) ω i (BA). This observation, although in different terms, was central to the original proof by Flanders [3]. Theorem 2.1. Let F be an algebraically closed field. Given A, B t F n m, the non-singular Jordan blocks of AB and BA have matching sizes, i.e., their Weyr characteristics are equal: (2.1) ω i (AB λi) =ω i (BA λi) for λ and all i. For the eigenvalue λ =, the Jordan forms of AB and BA have Weyr characteristics that satisfy (2.2) ω i 1 (AB) ω i (BA) ω i+1 (AB) for all i, which is equivalent to (2.3) σ i (AB) σ i (BA) 1 for all i. If P F n n and Q F m m satisfy ω i (P λi) = ω i (Q λi) for λ and ω i 1 (P ) ω i (Q) ω i+1 (P ), then there exist A, B t F n m such that P = AB and Q = BA. The equivalence of (2.2) and (2.3) is purely a combinatorial property of conjugate partitions (see Lemma 3.2). The Jordan block sizes are hence restricted to change by at most 1 for λ =. Taking Figure 2.1 as the Jordan structure of AB at λ =, Figure 2.2 is an admissible modification (by + and ) forba.

5 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June R.A. Lippert and G. Strang σ 1 =3 σ 2 =3 σ 3 =2 σ 4 =2 + σ 5 =1 + ω 1 =5 ω 2 =4 ω 3 =2 ω 4 = Fig If AB is nilpotent with Jordan structure J 4 J 4 J 2 J 1, then a permitted BA structure is J 3 J 3 J 2 J 2 J Main results. Our results are ultimately derived from the associativity of matrix multiplication. A typical example is B(AB AB) =(BA BA)B. Theorem 3.1. If A and B t are n m matrices over a field F, then for all i> ω i (AB λi) =ω i (BA λi) for λ F {} ω i (BA) ω i+1 (AB) (for λ =). Proof. (For λ ) For any polynomial p(x), p(ba)b = Bp(AB). Thus p(ab)v = implies p(ba)bv =. SinceBv = implies p(ab)v = p()v, we have dim Null(p(AB)) = dim Null(p(BA)) when p(). Hence ν i (AB λi) = ν i (BA λi) whenλ. (For λ = ) We define the following nullspaces for i : R i = {v F n : B(AB) i v =} R i = {v Fn :(AB) i v =} L i = {v F m : v t (BA) i =} L i = {v Fm : v t (BA) i B =} We see that, R i R i+1 and L i L i+1, and dim{r i+1} dim{r i } =dim{l i+1 } dim{l i }. Let v 1,...,v k R i+2 be a set of vectors that are linearly independent modulo R i+1. Thus k i=1 c iv i R i+1 only if c 1 = = c k =. Then the vectors

6 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June 29 Jordan forms of AB and BA 285 Fig A tableau representing the Jordan structure σ i =(1, 1, 7, 4, 3, 3, 1, 1, 1,,...), with Weyr characteristic ω i =(9, 6, 6, 4, 3, 3, 3, 2, 2, 2,,...). ABv 1,...,ABv k R i+1 are linearly independent modulo R i.thus, dim{r i+1 /R i} dim{r i+2 /R i+1}. I f v 1,...,v k L i+2 is a set of vectors, linearly independent modulo L i+1, then the vectors (BA) t v 1,...,(BA) t v k L i+1 are linearly independent modulo L i.thus,dim{l i+1 /L i} dim{l i+2 /L i+1}. Noticethat dim{r i+2 /R i+1} = ν i+2 (AB) dim{r i+1 } dim{l i+2/l i+1 } =dim{l i+2} ν i+1 (BA). Then dim{r i+1 /R i} dim{r i+2 /R i+1} implies dim{r i+2 } dim{r i+1 } ν i+2 (AB) ν i+1 (AB) and dim{l i+1 /L i} dim{l i+2 /L i+1} implies ν i+1 (BA) ν i (BA) dim{l i+2 } dim{l i+1 }. Therefore, ω i+1 (BA) ω i+2 (AB), since ω i+1 = ν i+1 ν i. The first part of Theorem 3.1 says that the Jordan structures of AB and BA for λ are identical, if F is algebraically closed. For a general field, the results can be adapted to show that the elementary divisors of AB and BA, that do not have zero as a root, are the same. An illustration is helpful in understanding the constraints implied by the second part, ω i 1 (AB) ω i (BA) ω i+1 (AB). Suppose the tableau in Figure 3.1 represents the Jordan form of AB at λ =. Theorem 3.1 constrains the tableau of the Jordan form of BA at λ = to be that of AB plus or minus the areas covered by the circles of Figure 3.2. The constraints on Weyr characteristics are equivalent to constraining the block sizes of the Jordan forms of AB and BA to differ by no more than 1. Although this

7 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June R.A. Lippert and G. Strang Fig Given AB (boxes), Theorem 3.1 imposes these constraints on the Weyr characteristic of BA (a circle can be added or subtracted from each row of the tableau): ω 1 6, 9 ω 2 6, 6 ω 3 4, 6 ω 4 3, 4 ω 5 3,ω 6 =3, 3 ω 7 2, 3 ω 8 2,ω 9 =2, 2 ω 9, 2 ω 1. equivalence is not hard to see [3] from Figure 3.1, it warrants a short proof. Taking d = 1, Lemma 3.2 establishes the equivalence of (2.2) and (2.3). Lemma 3.2. Let p 1 p 2 and p 1 p 2 be partitions of n and n with conjugate partitions q 1 q 2 and q 1 q 2.Letd N. Then q i q i+d and q i q i+d for all i> if and only if p i p i d for all i>. Proof. Ifp i >d,thenq p i>q pi+1 by the conjugacy conditions. By hypothesis, i q p i d q p >q pi+1 and thus p i d<p i +1 since q j is monotonically decreasing in j. i Thus p i p i + d (trivially true when p i d). By a symmetric argument (switching primed and unprimed), we have p i p i + d. Conversely, if q i+d >, then p q i+d p qi+d d (i + d) d = i>p q i +1, the first inequality by hypothesis and the next two by the conjugacy conditions. Since p j is monotonically decreasing, we have q i+d <q i + 1, and thus q i+d q i for all i> (trivially true when q i+d = ). A symmetric argument gives q i+d q i. What remains is to show that the constraints in Theorem 3.1 are exhaustive; we can construct matrices A, B that realize all the possibilities of the theorem. Here we find it easier to use the traditional Segre characteristic of block sizes σ i : Theorem 3.3. Let σ 1 σ 2 and σ 1 σ 2 be partitions of n and m respectively. If σ i σ i 1, then there exist n m matrices A and Bt such that σ j (AB) =σ j and σ j (BA) =σ j.

8 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June 29 Jordan forms of AB and BA 287 Proof. Foreachj such that σ j and σ j 1, we construct σ j σ j matrices A j and B t j such that A jb j = J σj () and B j A j = J σ j () according to these three cases: 1. σ j = σ j :seta j = J σj () and B j = I σj, [ ] 2. σ j +1=σ j :seta Iσj j =[ I σj ]andb j =, [ ] 3. σ j = σ j +1: seta Iσ j j = and B j =[ I σ j ]. This defines k =min{ω 1 (AB),ω 1 (BA)} matrix pairs (A j,b j ). Consider {σ j } as a partition for n rows and { σ j} as a partition for m columns. Construct the block diagonal matrix A =diag(a 1,...,A k,,...,) with zeros filling any remaining lower right part. Then with partitions { σ j} for m rows and {σj } for n columns let B = diag(b 1,...,B k,,...,). The final construction merely stitches together a singular piece with a nonsingular piece. Corollary 3.4. Let P F n n and Q F m m have Segre characteristics σi λ and σ i λ for each eigenvalue λ, i.e. P J σ λ i (λ) and Q J σ λ i (λ). λ F i> λ F i> If σi λ = σ i λ for all λ and σi σ i 1, then there exist matrices A and Bt in F n m such that P = AB and Q = BA. Proof. I f P = X 1 PX and Q = Y 1 QY are in canonical form with P = Ã B and Q = BÃ, then setting A = XÃY 1 and B = Y BX 1,wehaveP = AB and Q = BA. Hence we take P and Q to be in canonical form. Let M = λ i> J σ i (λ), i.e., M is a (non-singular) k k matrix in Jordan canonical form with Segre characteristic σi λ,wherek = λ i σλ i. Let A and B be the A and B matrices from Theorem 3.3 with σ i = σi and σ i = σ i. Then A = M A and B = I k B. Acknowledgment. We thank Roger Horn for pointing us to the Flanders paper and others, and for his encouragement. REFERENCES [1] James W. Demmel and Alan Edelman. The dimension of matrices (matrix pencils) with given Jordan (Kronecker) canonical forms. Linear Algebra Appl., 23:61 87, [2] Alan Edelman, Erik Elmroth, and Bo Kågström. A geometric approach to perturbation theory of matrices and matrix pencils. part II: A stratification-enhanced staircase algorithm. SIAM J. Matrix Anal. Appl., 2(3): , 1999.

9 Electronic Journal of Linear Algebra ISSN Volume 18, pp , June R.A. Lippert and G. Strang [3] Harley Flanders. Elementary divisors of AB and BA. Proc. Amer. Math. Soc., 2(6): , [4] W. V. Parker and B. E. Mitchell. Elementary divisors of certain matrices. Duke Math. J., 19(3): , [5] Helene Shapiro. The Weyr characteristic. Amer. Math. Monthly, 16(1): , [6] Robert C. Thompson. On the matrices AB and BA. Linear Algebra Appl., 1:43 58, 1968.