Five Theorems in Matrix Analysis, with Applications

f (AB), f (BA) Symmetr’n f(Jordan block) Sign function

Nick Higham School of Mathematics The University of Manchester

[email protected] http://www.ma.man.ac.uk/~higham/

Dundee (EMS)—March 17, 2006

Nick Higham Matrix Analysis 1 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Outline

f (AB) and f (BA) WMFME Λ(AB) and Λ(BA) f (αI + AB)

Symmetrization

Jordan Structure of f (A)

Matrix Sign Identities

Nick Higham Matrix Analysis 2 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) f (AB) and f (BA) For A, B Cn×n, AB = BA. ∈ 6 How are AB and BA related? How are f (AB) and f (BA) related? Same question if A Cm×n, B Cn×m. ∈ ∈ Generalize to f (αIm + AB) and f (αIn + BA).

Nick Higham Matrix Analysis 3 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Sherman–Morrison–Woodbury Formula If U, V Cn×p and I + V ∗A−1U is nonsingular then ∈ p (A + UV ∗)−1 = A−1 A−1U(I + V ∗A−1U)−1V ∗A−1. − p Obtained, using A + UV ∗ = A(I + A−1U V ∗), from its simpler version ·

A Cm×n (I + AB)−1 = I A(I + BA)−1B ∈ m − n B Cn×m ∈

Nick Higham Matrix Analysis 4 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) World’s Most Fundamental Matrix Equation

Nick Higham Matrix Analysis 5 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) World’s Most Fundamental Matrix Equation

(I + AB)A = A(I + BA), or

(AB)A = A(BA).

Nick Higham Matrix Analysis 5 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Application of WMFME

(AB)A = A(BA) (AB)2A = ABA(BA)= A(BA)2. ⇒ In general, for any poly p,

p(AB)A = Ap(BA).

◮ Does the same hold for arbitrary f ?

Nick Higham Matrix Analysis 6 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) AB and BA If A, B square and A nonsingular, WMFME implies AB = A(BA)A−1, so Λ(AB)= Λ(BA).

Nick Higham Matrix Analysis 7 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) AB and BA If A, B square and A nonsingular, WMFME implies AB = A(BA)A−1, so Λ(AB)= Λ(BA). Theorem (Flanders, 1951) Let A Cm×n and B Cn×m. ∈ ∈ The nonzero eigenvalues of AB have the same Jordan structure as the nonzero eigenvalues of BA. Any zero eigenvalues appear in Jordan blocks of AB and BA differing in size by at most 1.

Nick Higham Matrix Analysis 7 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Putnam Problem 1990-A5 If A, B Cn×n does ABAB = 0 imply BABA = 0? ∈

Nick Higham Matrix Analysis 8 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Putnam Problem 1990-A5 If A, B Cn×n does ABAB = 0 imply BABA = 0? ∈ Yes for n 2; no for n > 2. ≤ 0 0 1 0 0 1 A =  0 0 0  , B =  1 0 0  . 0 1 0 0 0 0     0 0 1 (AB)2 = 0, (BA)2 =  0 0 0  . 0 0 0  

Nick Higham Matrix Analysis 8 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Tridiagonal Toeplitz Matrices

d e .  c d ..  Tn(c, d, e)= ...... e    c d    Eigenvalues known explicitly:

d + 2(ce)1/2 cos(kπ/(n + 1)), k = 1: n.

What about simple modiﬁcations of Tn, e.g. to the (1,1) and (n, n) elements?

Nick Higham Matrix Analysis 9 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Second Difference Matrix

2 1 − .  1 2 ..  Tn = − , ......  1   1− 2   −  1 1 − .  1 2 ..  − . . Tn = .. .. .  1   −  e  1 2 1   − 1− 1   − 

Nick Higham Matrix Analysis 10 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Second Difference Matrix (cont.) Deﬁne 1  1 1  − . × L = 1 .. R(n+1) n.  −  ∈  ...   1   1   −  T T Then Tn = L L, Tn+1 = LL . So Λ(T )= Λ(T ) 0 (Strang, 2005). n+1 e n ∪ { } Example:e n = 6; L = gallery(’triw’,n,-1,1)’; L = L(:,1:n-1), A = L*L’, B = L’*L

Nick Higham Matrix Analysis 11 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Deﬁnition of Matrix Function

Let A have distinct eigenvalues λ1,...,λs, and let ni be order of the largest Jordan block in which λi appears. Deﬁnition (Sylvester, 1883) f (A) := r(A), where r is the unique Hermite interpolating s polynomial of degree less than i=1 ni that satisﬁes P r (j)(λ )= f (j)(λ ), j = 0: n 1, i = 1: s. i i i −

Nick Higham Matrix Analysis 12 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) f (AB) and f (BA) Recall that for any polynomial p,

Ap(BA)= p(AB)A.

Nick Higham Matrix Analysis 13 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) f (AB) and f (BA) Recall that for any polynomial p,

Ap(BA)= p(AB)A.

Lemma Let A Cm×n and B Cn×m and let f (AB) and f (BA) be deﬁned.∈ Then ∈ Af (BA)= f (AB)A.

Nick Higham Matrix Analysis 13 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) f (AB) and f (BA) Recall that for any polynomial p,

Ap(BA)= p(AB)A.

Lemma Let A Cm×n and B Cn×m and let f (AB) and f (BA) be deﬁned.∈ Then ∈ Af (BA)= f (AB)A.

Proof. There is a single polynomial p such that f (AB)= p(AB) and f (BA)= p(BA). Hence

Af (BA)= Ap(BA)= p(AB)A = f (AB)A.

Nick Higham Matrix Analysis 13 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Special Case Take f (t)= t 1/2. When AB (and hence also BA) has no eigenvalues on R−,

A(BA)1/2 =(AB)1/2A.

Nick Higham Matrix Analysis 14 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Special Case Take f (t)= t 1/2. When AB (and hence also BA) has no eigenvalues on R−,

A(BA)1/2 =(AB)1/2A.

———

Useful, but Af (BA)= f (AB)A cannot be solved for f (BA) in terms of f (AB).

Nick Higham Matrix Analysis 14 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB)

Theorem (Harris 1993; H 2005) Let A Cm×n and B Cn×m, with m n, and assume BA is nonsingular.∈ Then∈ ≥

f (αI + AB)= f (α)I + A(BA)−1 f (αI + BA) f (α)I B. m m n − n

Nick Higham Matrix Analysis 15 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB)

Theorem (Harris 1993; H 2005) Let A Cm×n and B Cn×m, with m n, and assume BA is nonsingular.∈ Then∈ ≥

f (αI + AB)= f (α)I + A(BA)−1 f (αI + BA) f (α)I B. m m n − n Proof. Deﬁne g(X)= X −1 f (αI + X) f (αI) . − Then f (αI + X)= f (α)I + Xg (X). Hence, using the lemma,

f (αIm + AB)= f (α)Im + ABg(AB)

= f (α)Im + Ag(BA)B = f (α)I + A(BA)−1 f (αI + BA) f (α)I B. m n − n

Nick Higham Matrix Analysis 15 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Example: Rank 2 Perturbation of I Consider f (αI + uv ∗ + xy ∗), where u, v, x, y Cn. Write n ∈ ∗ ∗ ∗ v uv + xy =[ u x ] ∗ AB. y ≡

Then ∗ ∗ v u v x 2×2 C := BA = ∗ ∗ C . y u y x ∈

∗ ∗ f (αIn + uv + xy )= f (α)In + ∗ −1 v [ u x ] C f (αI + C) f (α)I ∗ 2 − 2 y

Nick Higham Matrix Analysis 16 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function WMFME Λ(AB) and Λ(BA) f (αI + AB) Example: Rank 2 Perturbation of I Consider f (αI + uv ∗ + xy ∗), where u, v, x, y Cn. Write n ∈ ∗ ∗ ∗ v uv + xy =[ u x ] ∗ AB. y ≡

Then ∗ ∗ v u v x 2×2 C := BA = ∗ ∗ C . y u y x ∈

∗ ∗ f (αIn + uv + xy )= f (α)In + ∗ −1 v [ u x ] C f (αI + C) f (α)I ∗ 2 − 2 y

For A C2×2, f (A)= f (λ )I + f [λ ,λ ](A λ I). ∈ 1 1 2 − 1

Nick Higham Matrix Analysis 16 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Outline

f (AB) and f (BA) WMFME Λ(AB) and Λ(BA) f (αI + AB)

Symmetrization

Jordan Structure of f (A)

Matrix Sign Identities

Nick Higham Matrix Analysis 17 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Symmetrization Theorem (Frobenius, 1910) For any A Fn×n (F = R or C) there exist symmetric S , S Fn×∈n, either one of which can be taken 1 2 ∈ nonsingular, such that A = S1S2.

Nick Higham Matrix Analysis 18 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Symmetrization Theorem (Frobenius, 1910) For any A Fn×n (F = R or C) there exist symmetric S , S Fn×∈n, either one of which can be taken 1 2 ∈ nonsingular, such that A = S1S2.

Implication The generalized eigenproblem Ax = λBx with symmetric A and B has no special eigenproperties: equivalent to Cx := B−1Ax = λx, with C arbitrary.

Nick Higham Matrix Analysis 18 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Proof Rational canonical form says A is similar to a direct sum −1 −1 of companion matrices: A = X CX. But S1 C = S2:

0 0 1 β2 β1 β0 0 1 0  0 1 β2   1 0 0  =  1 β2 0  . 1 β −β 0 1 0 0− 0 β  − 2 − 1     0  Then A = X −1S S X = X −1S X −T X T S X S S . 1 2 1 · 2 ≡ 1 2 e e

Nick Higham Matrix Analysis 19 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Proof Rational canonical form says A is similar to a direct sum −1 −1 of companion matrices: A = X CX. But S1 C = S2:

0 0 1 β2 β1 β0 0 1 0  0 1 β2   1 0 0  =  1 β2 0  . 1 β −β 0 1 0 0− 0 β  − 2 − 1     0  Then A = X −1S S X = X −1S X −T X T S X S S . 1 2 1 · 2 ≡ 1 2 e e Theorem For any A Fn×n (F = R or C) there exists a nonsingular symmetric∈ S such that A = S−1AT S.

Nick Higham Matrix Analysis 19 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Application to Polynomial Zero-Finding Lancaster (1961) takes companion linearization λI C for scalar poly p(t)= a t k + + a t + a : − k ··· 1 0 a − /a a − /a . . . a /a − k 1 k − k 2 k − 0 k  1 0 . . . 0  C = ......    1 0   −1  We can write C = S1 S2 with S1, S2 symm. So ◮ S (λI C)= λS S C is a symm. pencil. 1 − 1 − 1 ◮ Ditto S Cℓ−1(λI C)= λS Cℓ−1 S Cℓ for ℓ 1. 1 − 1 − 1 ≥ Lancaster takes a .. k . ak−1 S1 =  . . .  ......  a a − . . . a   k k 1 1 

Nick Higham Matrix Analysis 20 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Matrix Polynomial Case This construction generalizes immediately to matrix polynomials and provides block symmetric pencils λX + Y [X = X , i = j]. ij ji 6 ◮ What space of pencils is generated? ◮ What happens as ℓ increases?

◮ Is there anything special about this particular S1? ◮ How are ei’vecs of the pencils related to those of P?

Nick Higham Matrix Analysis 21 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Matrix Polynomial Case This construction generalizes immediately to matrix polynomials and provides block symmetric pencils λX + Y [X = X , i = j]. ij ji 6 ◮ What space of pencils is generated? ◮ What happens as ℓ increases?

◮ Is there anything special about this particular S1? ◮ How are ei’vecs of the pencils related to those of P?

Answered via a new theory of vector spaces of linearizations: H, D. S. Mackey, N. Mackey, Mehl, Mehrmann, Tisseur (2005)

Nick Higham Matrix Analysis 21 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Outline

f (AB) and f (BA) WMFME Λ(AB) and Λ(BA) f (αI + AB)

Symmetrization

Jordan Structure of f (A)

Matrix Sign Identities

Nick Higham Matrix Analysis 22 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Function of Jordan block A = Z diag(J ,..., J )Z −1 f (A)= Z diag(f (J ),..., f (J ))Z −1. 1 p ⇒ 1 p

λk 1 ..  λ .  × k Cmk mk Jk = . ,  .. 1  ∈    λk   

(mk −1) ′ f )(λk ) f (λk ) f (λk ) . . .  (mk 1)!  − .. . f (Jk )=  f (λk ) . .  .  .   .. f ′   (λk )   f (λ )   k 

Nick Higham Matrix Analysis 23 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function

Theorem Let A Cn×n with eigenvalues λ . ∈ k ′ 1 If f (λ ) = 0 then for every J(λ ) in A there is a Jordan k 6 k block of the same size in f (A) for f (λk ). ′ ′′ − 2 Let f (λ )= f (λ )= = f (ℓ 1)(λ )= 0 but f (ℓ)(λ ) = 0, k k ··· k k 6 where ℓ 2, and consider J(λ ) of size r in A. ≥ k (i) If ℓ r, J(λ ) splits into r 1 1 Jordan blocks for ≥ k × f (λk ) in f (A). (ii) If ℓ r 1, J(λ ) splits into Jordan blocks for f (λ ) ≤ − k k in f (A) as follows: ℓ q Jordan blocks of size p, • − q Jordan blocks of size p + 1, • where r = ℓp + q with 0 q ℓ 1,p > 0. ≤ ≤ − Nick Higham Matrix Analysis 24 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function

Nick Higham Matrix Analysis 25 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Application: Matrix Logarithm Find all solutions to eX = A. −1 −1 Let A have JCF A = Z diag(Jk (λk ))Z = ZJZ . Since d ex = 0 , X has Jordan form dx 6 JX = diag(Jk (µk )), where exp(µk )= λk and hence µk = log λk + 2jk πi.

Now consider L = diag(Lk ), where L Lk = log(Jk (λk )) + 2jk πiI. Then e = J , so by same argument as above, L has Jordan form JX , i.e., X = WLW −1, some W .

Now consider L = diag(Lk ), where L Lk = log(Jk (λk )) + 2jk πiI. Then e = J , so by same argument as above, L has Jordan form JX , i.e., X = WLW −1, some W . But eX = A implies WJW −1 = WeLW −1 = ZJZ −1, or (Z −1W )J = J(Z −1W ).

Nick Higham Matrix Analysis 25 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Application: Matrix Logarithm Theorem (Gantmacher, 1959)

n×n −1 Let A C be nonsing. with JCF A = Z diag(Jk (λk ))Z . All solutions∈ to eX = A are given by

(j1) (j2) (jp) −1 −1 X = ZUdiag(L1 , L2 ,..., Lp )U Z , where (jk ) Lk = log(Jk (λk )) + 2jk πiI,

log(Jk (λk )) is the principal logarithm, jk is an arbitrary integer, and U is an arbitrary nonsingular matrix commuting with J.

Nick Higham Matrix Analysis 26 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Outline

f (AB) and f (BA) WMFME Λ(AB) and Λ(BA) f (αI + AB)

Symmetrization

Jordan Structure of f (A)

Matrix Sign Identities

Nick Higham Matrix Analysis 27 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Matrix Sign Function For A Cn×n with Jordan canonical form ∈ J 0 − A = Z 1 Z 1, 0 J2

where λ(J ) open LHP, λ(J ) open RHP, 1 ∈ 2 ∈

I 0 − sign(A)= Z − Z 1. 0 I Introduced by Roberts (1971), who proposed Newton iter.

1 X = (X + X −1), X = A. k+1 2 k k 0

Xk converges quadratically to sign(A).

Nick Higham Matrix Analysis 28 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Matrix Sign Relations For nonsingular A Cn×n (Byers, 1984): ∈ 0 A 0 U sign ∗ = ∗ , A 0 U 0

where A = UH is the polar decomposition.

Nick Higham Matrix Analysis 29 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Matrix Sign Relations For nonsingular A Cn×n (Byers, 1984): ∈ 0 A 0 U sign ∗ = ∗ , A 0 U 0

where A = UH is the polar decomposition.

For A Cn×n with no eigenvalues on R− (H, 1997): ∈ 0 A 0 A1/2 sign = − . I 0 A 1/2 0

Nick Higham Matrix Analysis 29 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function More General Matrix Sign Relation Theorem (H, Mackey, Mackey, Tisseur, 2005) Let A, B Cn×n and suppose AB has no eigenvalues on R−. Then∈ 0 A 0 C sign = − , B 0 C 1 0

where C = A(BA)−1/2.

Nick Higham Matrix Analysis 30 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function

0 A Proof. P = B 0 has no pure imaginary ei’vals. Hence −1/2 − 0 A AB 0 sign(P)= P(P2) 1/2 = B 0 0 BA 0 A (AB)−1/2 0 = − B 0 0 (BA) 1/2 0 A(BA)−1/2 0 C = − =: . B(AB) 1/2 0 D 0

Now

0 C 2 CD 0 I =(sign(P))2 = = , D 0 0 DC

so D = C−1.

Nick Higham Matrix Analysis 31 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Puzzle Proof of previous theorem shows that

−1 A(BA)−1/2 = B(AB)−1/2 =(AB)1/2B−1. Why do we have equality?

Nick Higham Matrix Analysis 32 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Puzzle Proof of previous theorem shows that

−1 A(BA)−1/2 = B(AB)−1/2 =(AB)1/2B−1. Why do we have equality?

Recall Af (BA)= f (AB)A . Now

A(BA)−1/2 B(AB)−1/2 =(AB)−1/2A B(AB)−1/2 · · =(AB)−1/2AB(AB)−1/2 = I.

Nick Higham Matrix Analysis 32 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Application: Matrix Iterations Apply any iteration for the matrix sign function to

0 A 0 A ∗ or A 0 I 0

and read off from the (1,2) block an iteration for polar factor U or A1/2. Applying the lemma to

0 A A⋆ 0

can derive new iterations for the generalized polar decomposition this way (HMMT, 2005).

Nick Higham Matrix Analysis 33 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Summary λ(AB) vs. λ(BA) : Flanders (1951).

f (αIm + AB) : Harris (1993), H (2005).

A = S1S2 : Frobenius (1910). Jordan structure of f (J) .

0 A sign( B 0 ) : H, Mackey, Mackey, Tisseur (2005).

Nick Higham Matrix Analysis 34 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography I

A. J. Bosch. Note on the factorization of a square matrix into two Hermitian or symmetric matrices. SIAM Rev., 29(3):463–468, 1987. Harley Flanders. Elementary divisors of AB and BA. Proc. Amer. Math. Soc., 2(6):871–874, 1951. F. R. Gantmacher. The Theory of Matrices, volume two. Chelsea, New York, 1959.

Nick Higham Matrix Analysis 35 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography II P. R. Halmos. Bad products of good matrices. Linear and Multlinear Algebra, 29:1–20, 1991. Lawrence A. Harris. Computation of functions of certain operator matrices. Linear Algebra Appl., 194:31–34, 1993. Nicholas J. Higham. Functions of a Matrix: Theory and Computation. Book in preparation.

Nick Higham Matrix Analysis 36 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography III Nicholas J. Higham, D. Steven Mackey, Niloufer Mackey, and Françoise Tisseur. Functions preserving matrix groups and iterations for the matrix square root. SIAM J. Matrix Anal. Appl., 26(3):849–877, 2005. Nicholas J. Higham, D. Steven Mackey, Niloufer Mackey, and Françoise Tisseur. Symmetric linearizations for matrix polynomials. MIMS EPrint 2005.25, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, November 2005. Submitted to SIAM J. Matrix Anal. Appl.

Nick Higham Matrix Analysis 37 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography IV Roger A. Horn and Dennis I. Merino. Contragredient equivalence: A canonical form and some applications. Linear Algebra Appl., 214:43–92, 1995. Charles R. Johnson and Eric Schreiner. The relationship between AB and BA. Amer. Math. Monthly, 103(7):578–582, 1996. Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson. The ﬁfty-ﬁrst William Lowell Putnam mathematical competition. Amer. Math. Monthly, 98(8):719–727, 1991.

Nick Higham Matrix Analysis 38 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography V Peter Lancaster. Symmetric transformations of the companion matrix. NABLA: Bulletin of the Malayan Math. Soc., 8:146–148, 1961. Heydar Radjavi. Products of Hermitian matrices and symmetries. Proc. Amer. Math. Soc., 21(2):369–372, 1969. O. Taussky. The role of symmetric matrices in the study of general matrices. Linear Algebra Appl., 5:147–154, 1972.

Nick Higham Matrix Analysis 39 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function Bibliography VI Olga Taussky and Hans Zassenhaus. On the similarity transformation between a matrix and its transpose. Paciﬁc J. Math., 9:893–896, 1959. R. C. Thompson. On the matrices AB and BA. Linear Algebra Appl., 1:43–58, 1968.

Nick Higham Matrix Analysis 40 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function World’s Most Fundamental Matrix Equation

(I + AB)−1 = I A(I + BA)−1B. −

I = I + AB (I + AB)A(I + BA)−1B − = I + AB A(I + BA)(I + BA)−1B − = I + AB AB − = I √

Nick Higham Matrix Analysis 41 f (AB), f (BA) Symmetr’n f(Jordan block) Sign function World’s Most Fundamental Matrix Equation

(I + AB)−1 = I A(I + BA)−1B. −

I = I + AB (I + AB)A(I + BA)−1B − = I + AB A(I + BA)(I + BA)−1B − = I + AB AB − = I √

Key equation: (I + AB)A = A(I + BA), or

(AB)A = A(BA).

Nick Higham Matrix Analysis 41