Linear Algebra and its Applications

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1 Linear Algebra and its Applications 36 ( Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: The sum of orthogonal matrices Dennis I. Merino Department of Mathematics, Southeastern Louisiana University, Hammond, LA , USA ARTICLE INFO Article history: Received 1 October 010 Accepted 3 October 011 Availableonline3November011 Submitted by R.A. Brualdi Keywords: Sum of unitary matrices Sum of orthogonal matrices ABSTRACT Let F {R, C, H}. Let U n (F be the set of unitary matrices in M n (F, andleto n (F be the set of orthogonal matrices in M n (F. Suppose n. We show that every A M n (F can be written as a sum of matrices in U n (F and of matrices in O n (F. Let A M n (F be given and let k be the least integer that is a least upper bound of the singular values of A. When F = C, we show that A can be written as a sum of k matrices from U n (F. WhenF = R,weshow that if k 3, then A can be written as a sum of 6 orthogonal matrices; if k, we show that A can be written as a sum of k + orthogonal matrices. 011 Elsevier Inc. All rights reserved. 1. Introduction Let F = C (the set of complex numbers or F = R (the set of real numbers. Let n be a given positive integer. We let M n (F be the set of all n-by-n matrices with entries in F. WealsoletE ij M n (F be the matrix whose (i, j entry is 1 and all other entries are 0. We study the sums of unitary matrices and we also study the sums of orthogonal matrices. We determine which matrices (if any in M n (F can be written as a sum of unitary or orthogonal matrices. We note that the sum of unitary matrices in M n (C has been previously studied (see [5] and the references therein. Moreover, for A, B M n (C,sumsoftheformUAU +VBV, where U, V M n (C are unitary, have also been studied []. We let U n (F be the set of unitary matrices in M n (F and we let O n (F be the set of orthogonal matrices in M n (F. We begin with the following observation. Lemma 1. Let n be a given positive integer. Let G M n (F be a group under multiplication. Then A M n (F can be written as a sum of matrices in G if and only if for every Q, P G,thematrixQAPcan bewrittenasasumofmatricesing. address: dmerino@selu.edu /$ - see front matter 011 Elsevier Inc. All rights reserved. doi: /j.laa

2 D.I. Merino / Linear Algebra and its Applications 36 ( Notice that both U n (F and O n (F are groups under multiplication. Let α F be given. Then Lemma 1 guarantees that for each Q G, we have that αq can be written as a sum of matrices from G if and only if αi can be written as a sum of matrices from G. Lemma. Let n be a given integer. Let G M n (F be a group under multiplication. Suppose that G contains K diag(1, 1,..., 1 and the permutation matrices. Then every A M n (F can be written as a sum of matrices in G if and only if for each α F, αi canbewrittenasasumofmatricesing. Proof. The forward implication is trivial. For the other direction, suppose that for each α F, αi can be written as a sum of matrices in G. Now,K G so that for each α F, Lemma 1 guarantees that αk can also be written as a sum of matrices in G. It follows that αe 11 = α I + α K can be written as a sum of matrices in G. Now,foreach1 i, j n, notice that E ij = PE 11 Q for some permutation matrices P and Q, and that P, Q G. Therefore, if A M n (F, thena can be written as a sum of matrices in G, as A = [ ] a ij = i,j a ij E ij.. Sum of orthogonal matrices The only matrices in O 1 (F are ±1. Hence, not every element of M 1 (F canbewrittenasasum of elements in O 1 (F. In fact, only the integers can be written as a sum of elements of O 1 (F..1. The case F = C Notice that U 1 (C = { e iθ : θ R }. Set C { e iθ + e iβ : θ,β R }. Ifθ,β R are given, then e iθ + e iβ. Hence, C A {z C : z }. We show that A C. Let 0 r be given. Set β = θ, and choose θ so that cos θ = r. Thene iθ + e iθ = cos θ = r. Ifz = re iδ,then choose β = θ + δ, and choose θ so that cos (θ δ = r. Let k be an integer. Set C k { } k j=1 eiθ j : θ j R for j=1,..., k and set A k {z C : z k}. We show that for each k, wehavea k = C k. First, notice that for each k, wehavec k A k. We now show that A k C k. If z = re iβ,with r,β R and r 0, then e iθ 1 + +e iθ k = re iβ if and only if e i(θ 1 β + +e i(θ k β = r. (1 Hence, z = re iβ C k if and only if r C k. Forθ 1,..., θ k R, setf k (θ 1,..., θ k e iθ 1 + +e iθ k. The case k = has already been shown. Let k = 3, and suppose 0 r 3. Set θ 3 = 0 and set θ 1 = θ = θ. Then, f 3 (θ 1,θ,θ 3 = 1 + cos θ, and θ may be chosen so that 0 r 1 + cos θ 3. We use mathematical induction to show the general case. The base cases k = and k = 3have already been shown. Assume that k > 3 and suppose that C k = A k. Consider f k+1 (θ 1,..., θ k,θ k+1 e iθ 1 + +e iθ k +e iθ k+1.letz = re iβ be given with 0 r k+1. We show that r C k+1. First, we show that A C k+1.ifkis even, choose θ 3 = =θ k 1 = 0 and θ = =θ k = π. Then f k+1 (θ 1,..., θ k,θ k+1 = e iθ 1 + e iθ. If k is odd, choose θ = = θ k 1 = 0 and θ 5 = = θ k = π. Thenf k+1 (θ 1,..., θ k,θ k+1 e iθ 1 + e iθ + e iθ 3. In both cases, notice that A C k+1. Hence, we may assume further that r 1; that is, we need to show that r C k+1 for 1 r k + 1. Choose θ k+1 = 0, so that f k+1 (θ 1,..., θ k,θ k+1 = f k (θ 1,..., θ k + 1. Now, by our inductive hypothesis, the equation f k (θ 1,..., θ k + 1 = r has a solution since 0 r 1 k. Lemma 3. Let k be a given integer. Let A k {z C : z k} and let C k { k j=1 eiθ j : θ j R for j = 1,..., k}. Then A k = C k.

3 196 D.I. Merino / Linear Algebra and its Applications 36 ( The case U n (C Let α C be given. Then there exist an integer k and θ 1,..., θ k R such that α = f k (θ 1,..., θ k. Now, notice that αi = f k (θ 1,..., θ k I = e iθ 1 I + +e iθ k I is a sum of matrices in U n (C. When n = 1, every α C can be written as a sum of elements of U 1 (C. When n, Lemma guarantees that every A M n (C can be written as a sum of matrices in U n (C. Lemma. Let n be a given positive integer. Then every A M n (C can be written as a sum of matrices in U n (C. Let A M n (C be given. We look at the number of matrices that make up the sum A. Let α C be given. If α k for some positive integer k, thenα A k. Moreover, α A m for every integer m k. For any such m, Lemma 3 guarantees that there exist θ 1,..., θ m R such that α = e iθ 1 + +e iθ m. However, if α > k, thenα / A k and α cannot be written as a sum of k elements of U 1 (C. Write A = U V (the singular value decomposition of A, see[1, Theorem 7.3.5] or [, Theorem 3.1.1], where U, V M n (C are unitary and = diag(σ 1,..., σ n with σ 1 σ n 0. Let k be the least integer such that σ 1 k. Suppose that k. Then, for each l, wehaveσ l A k. Moreover, σ 1 / A k 1. Hence, A cannot be written as a sum of k 1 unitary matrices. However, for each l, wehaveσ l = e iθ l1 + +e iθ lk, where each θ l1,..., θ lk R. For each t = 1,..., k, setu t = diag (e iθ 1t,..., e iθ nt. Then U t M n (C is unitary and k t=1 U t =. Hence, A can be written as a sum of k unitary matrices. Suppose that k = 1. If σ n = 1, then = I and A is unitary. If σ n = 1, then for each l, wehave σ l A, and A can be written as a sum of two unitary matrices. We summarize our results. Theorem 5. Let A M n (C be given. Let k be the least (positive integer so that there exist U 1,..., U k U n (C satisfying U 1 + +U k = A. 1. If A is unitary, then k = 1.. If A is not unitary and σ 1 (A, thenk=. 3. If m is an integer such that m <σ 1 (A m + 1, thenk= m + 1. For positive integers m k, wehavea k A m. Hence, every U U n (C canbewrittenasasum of two or more elements of U n (C. It follows that every A M n (C that can be written as a sum of k elements of U n (C can be written as a sum of m elements of U n (C..1.. The case O n (C Let n =. Let α, β C be given. Set A (α, β α β. ( β α Choose β such that α + β = 1 and notice that A (±α, ±β O (C. Set A 1 A (α, β and set A A (α, β. Then A 1 + A = αi. Lemma guarantees that every A M (C can be written as a sum of matrices from O (C. We look at the case when n = 3. Let α, β F be given. Set B (α, β 0 αβ, (3 0 β α

4 D.I. Merino / Linear Algebra and its Applications 36 ( set α 0 β C (α, β 01 0, ( β 0 α and set αβ 0 D (α, β β α 0. ( Choose β so that α + β = 1. Then, B (±α, ±β, C (±α, ±β, and D (±α, ±β are all elements of O 3 (C. Set B 1 B (α, β, setb B ( α, β, setc 1 C (α, β, setc C ( α, β, set D 1 D (α, β, and set D D ( α, β. Then, B 1 B + C 1 C + D 1 D = αi 3. Lemma now guarantees that every A M 3 (C can be written as a sum of matrices in O 3 (C. Let n = m be a positive even integer, and let δ C be given. Choose A 1, A O (C so that A 1 + A = δi. Set E 1 = A 1 A 1 (m copies and set E = A A (m copies. Then E 1, E O m (C, and E 1 + E = δi m. Let n = m be an odd integer, and let δ C be given. Choose A 1, A O (C so that A 1 +A = δi. Also, choose B 1, B, C 1, C, D 1, D O 3 (C such that B 1 B +C 1 C +D 1 D = δi 3. Set E 1 = A 1 A 1 B 1 (m 1 copies of A 1, set E = A A B (m 1 copies of A, set E 3 = I m C 1,setE = I m C,setE 5 = I m D 1, and set E 6 = I m D. Then each E j O m+1 (C, and E 1 + +E 6 = δi m+1. Hence, for every α C and for every integer n, αi can be written as a sum of matrices from O n (C. Lemma guarantees that every A M n (C can be written as a sum of matrices from O n (C. Theorem 6. Let n be a given integer. Then every A M n (C can be written as a sum of matrices from O n (C. Suppose that A = Q 1 + Q, where Q 1, Q O n (C. Then one checks that AA T = Q 1 A T AQ1 T,so that AA T and A T A are similar. Theorem 13 of [3] ensures that A = QS, where Q is orthogonal and S is symmetric (or that A has a QS decomposition. Suppose now that has a QS decomposition. Is it true that A can be written as a sum of two (complex orthogonal matrices? Take the case n = 1, and notice that every A M n (C is a scalar and has a QS decomposition. However, only the integers can be written as a sum of orthogonal matrices in this case. Lemma 7. Let an integer n and 0 = α C be given. If αi = Q + V is a sum of two matrices from O n (C, then there exists a skew-symmetric D M n (C such that Q = α I + D, V = α I D, and DD T = ( 1 α I. Conversely, if there exists a skew-symmetric D M n (C such that DD T = ( 1 α I, then Q α I + DandV α I DareinO n (C and Q + V = αi. Proof. Let an integer n and 0 = α C be given. Suppose that αi M n (C can be written as a sum of two orthogonal matrices, say, αi = Q + V. Write Q = [ ] a ij = [q1... q n ] and V = [ ] b ij = [v 1... v n ]. Then, b ij = a ij for i = j. Now, for each i = 1,..., n, wehave n j=1 a ij = q T i q i = 1 = v T i v i = n j=1 b ij = b + n ii j =i,j=1 a ij. Hence, b ii =±a ii. Because Q + V = αi and α = 0, we have b ii = a ii = α. Set D = [ ] d ij,withdij = a ij if i = j, and d ii = 0, so that Q = α I + D and V = α I D. Now, since Q and V are orthogonal, we have QQ T = α I + α ( D + D T + DD T = I (6

5 196 D.I. Merino / Linear Algebra and its Applications 36 ( and VV T = α I α ( D + D T + DD T = I. (7 Subtracting Eq. (7 fromeq.(6, we get D = D T, so that D is skew-symmetric. Moreover, DD T = (1 α I. For the converse, suppose that D M n (C is skew-symmetric and satisfies DD T = ( 1 α I. Set Q α I + D and set V α I D. Then one checks that both Q and V are orthogonal matrices and Q + V = αi. If α = 0, then for any orthogonal Q, notice that αi = Q + ( Q is a sum of two orthogonal matrices. Let n = and α = 0. Set β 1 α (either square root. Then D 0 β is β 0 skew-symmetric and satisfies DD T = ( 1 α I. Lemma 7 guarantees that αi can be written as a sum of two orthogonal matrices. If n = k and α = 0, set E D D (k copies and notice that I. Hence,ifn = k and if α is a scalar, then αi can E is skew-symmetric and satisfies EE T = ( 1 α be written as a sum of two orthogonal matrices. Theorem 8. Let n be a given positive integer. For each α C and each orthogonal Q M n (C, αq can be written as a sum of two orthogonal matrices. Let an integer n begiven. Ifα {, 0, }, then one checks that αi M n (C can be written as a sum of two orthogonal matrices. Theorem 9. Let α C and let a positive integer n be given. Then αi M n+1 (C can be written as a sum of two matrices from O n (C if and only if α {, 0, }. Proof. For the forward implication, let α C and let a positive integer n be given. Suppose that αi M n+1 (C can be written as a sum of two orthogonal matrices. Then α = 0orα = 0. If α = 0, then α {, 0, }. Ifα = 0, we show that α =±. Lemma 7 guarantees that there exists a skew-symmetric D M n (C satisfying DD T = ( 1 α D is singular. Hence, DD T is singular and α =±. The backward implication can be shown by direct computation... The case F = R I. Now,sincen is odd, the skew-symmetric Notice that O n (R = U n (R. Whenn = 1, only the integers can be written as a sum of elements of O 1 (R. Suppose that n =. We mimic the computations done in the case when F = C. Let θ R be given, set α = cos θ, and set β = sin θ. Then A (α, β in Eq. ( isanelementofo (R. Moreover, A 1 + A = cos θi.now,foreveryδ R,thereexistapositiveintegerm and θ R such that m cos θ = δ. We conclude that every A M (R can be written as a sum of an even number of matrices from O (R. When n = 3, we again mimic the computations done in the case when F = C using α = cos θ and β = sin θ to show that for every δ R,thematrixδI 3 can be written as a sum of an even number of matrices from O 3 (R. Let n be a given integer. If n = k is even, then write δi k = δi δi (k copies, and note that each δi can be written as a sum of an even number of matrices from O (R. Ifn = k + 1 is odd, then write δi k+1 = δi n δi 3. Now,δI n can be written as a sum of an even number of matrices from O n (R and δi 3 can be written as a sum of an even number of matrices from O 3 (R. We conclude that δi k+1 can be written as a sum of an even number of matrices from O k+1 (R.

6 D.I. Merino / Linear Algebra and its Applications 36 ( Hence, Lemma guarantees that for every integer n, every A M n (R can be written as a sum of matrices from O n (R. Theorem 10. Let n be a given integer. Every A M n (R can be written as a sum of matrices from O n (R = U n (R. Let n be a given integer and let U U n (R be given. Then U U n (C O n (C, that is, a real orthogonal matrix is both complex unitary and complex orthogonal. Hence, an A M n (R, which is a sumofmatricesinu n (R is both a sum of complex unitary matrices and a sum of complex orthogonal matrices. Thus, the restrictions on these cases apply. If k is a positive integer such that σ 1 (A > k, then A cannot be written as a sum of k real orthogonal matrices. Let m be a positive integer. Theorem 9 guarantees that I M m+1 (C cannot be written as a sum of two matrices in O m+1 (C. Now,I cannot be written as a sum of two matrices from O m+1 (R O m+1 (C. In general, if α/ {, 0, } and if Q O m+1 (R, thenαq cannot be written as a sum of two matrices from O m+1 (R. Let n be a given integer, and let A M n (R be given. We now look at the matrices in O n (R that make up the sum A. Definition 11. Let θ R be given. We define A (θ cos θ sin θ and B (θ cos θ sin θ. (8 sin θ cos θ sin θ cos θ Set K B (0 and notice that A (0 = I. Let 0 r, s R be given, and let k be an integer. If r, s k, then Lemma 3 and taking the real and imaginary parts of the equation e iθ 1 + +e iθ k = α (9 show that there exist θ 1,..., θ k R such that A (θ 1 + +A (θ k = ri. Moreover, there exist β 1,..., β k R such that B (β 1 + +B (β k = sk. Theorem 1. Let a positive integer n and let A M n (R be given. Suppose that k is an integer such that σ 1 (A k. Then A can be written as a sum of k matricesino n (R. Moreover, for every integer m k, the matrix A can be written as a sum of m matrices in O n (R. Proof. Let A = U V be a singular value decomposition of A. Then Lemma 1 guarantees that we only need to concern ourselves with. For n = 1, notice that diag(σ 1,σ = si + tk, where s = 1 (σ 1 + σ and t = 1 (σ 1 σ. Now, 0 t s k. Notice that si and tk can each be written as a sum of k orthogonal matrices. Moreover, for each integer p k, notice that si can be written as a sum of p orthogonal matrices. Hence, si + tk can be written as a sum of p + k orthogonal matrices. For n > 1, write = diag(σ 1,σ,..., σ n 1,σ n = diag(σ 1,σ diag(σ n 1,σ n.notice now that for each j = 1,..., n,diag ( σ j 1,σ j can be written as a sum of k orthogonal matrices, say P j1,..., P j(k. For each l = 1,..., k, setq l P 1l P nl, and notice that = Q 1 + +Q k. Finally, notice that for each integer m k and for each j = 1,..., n, the matrix diag ( σ j 1,σ j is also a sum of m orthogonal matrices. Consider C 0 diag(b, a with real numbers b a 0. If b, then Theorem 1 ensures that C 0 can be written as a sum of real orthogonal matrices. Moreover, for each integer t, C 0 can be written as a sum of t real orthogonal matrices. Suppose that b 3. If 0 b, then Theorem 1 guarantees that C 0 can be written as a sum of real orthogonal matrices. Moreover, C 0 can also be written as a sum of 5 real orthogonal matrices. If < b 3, then we look at two cases: (i 0 a 1 and (ii 1 a 3. In the first case, set C 1 C 0 K. Then 0 b 1 and 0 a + 1 <. Notice now that for each integer t, C 1

7 1966 D.I. Merino / Linear Algebra and its Applications 36 ( can be written as a sum of t real orthogonal matrices. In the second case, set C 1 C 0 I. Then we have 0 a 1 b 1. Theorem 1 guarantees that for each integer t, C 1 can be written as a sum of t real orthogonal matrices. Hence, for each integer t 5, C 0 can be written as a sum of t real orthogonal matrices. We now use induction to show that if k is an integer satisfying b k, thenforeachinteger t k +, C 0 can be written as a sum of t real orthogonal matrices. Suppose that the claim is true for some integer k 3. We show that the claim is true when 0 b k + 1. If 0 b k, then our inductive hypothesis guarantees that for each integer t k +, C 0 can be written as a sum of t (and hence, also of t k + 3 real orthogonal matrices. If k < b k + 1, we take a look at two cases: (i 1 a k + 1 and (ii 0 a < 1. In case (i, set C 1 C 0 I ; and in case (ii, set C 1 C 0 K. We summarize our results. Lemma 13. Let C M (R be given. Suppose that k is an integer such that σ 1 (C k. Then for each integer t k +, C can be written as a sum of t matrices from U (R. Let A M n (R be given, and suppose that the singular values of A are σ 1 σ n 0. Set D diag(σ 1,..., σ n.writed = diag(σ 1,σ diag(σ n 1,σ n.letk be an integer such that σ 1 (A k. Then Lemma 13 guarantees that for each integer t k +, and for each j = 1,..., n, diag ( σ j 1,σ j canbewrittenasasumoft real orthogonal matrices. We conclude that for each integer t k +, A can be written as a sum of t real orthogonal matrices. Theorem 1. Let n be positive integer, and let A M n (R be given. Suppose that k is an integer such that σ 1 (A k. Then for each integer t k +, A can be written as a sum of t matrices in U n (R. Let A M 3 (R be given. Suppose that A = P Q, withp, Q O 3 (R and = diag(a, b, c with 0 c b a. If a =, then notice that diag(b, c can be written as a sum of four orthogonal matrices. One checks that can be written as a sum of four real orthogonal matrices. Suppose a <. If c = 0, then can be written as a sum of four orthogonal matrices. If 0 = c <, then choose θ so that cos θ = c. Notice that A (θ + A ( θ = cos θi.setu 1 = [1] A (θ and set U = [ 1] A ( θ. Then (U 1 + U = diag(a, b c, 0, which can be written as a sum of four real orthogonal matrices. Hence, A can be written as a sum of six real orthogonal matrices. Let n be a given positive integer and let A M n+1 (R. Let A=P Q be the singular value decomposition of A. Suppose that k is a positive integer such that σ 1 (A k. If k, then A can be written as a sum of at most six matrices in O n+1 (R. If k > and if σ 1 (A =k, wewrite = [k] diag(σ,σ 3 diag(σ n,σ n+1. Notice that each of the -by- matrices can be written as a sum of k+ (or more orthogonal matrices. Hence, canbewrittenasasumofk+matricesfromo n+1 (R. Suppose that σ 1 < k. Ifk = 3, choose θ so that σ 1 cos θ =. For j = 3,..., n + 1, set d j = 1; if σ j, then set e j = 1, and if σ j <, then set e j = 1. Set D = diag(d 3,..., d n+1 and set E = diag(e 3,..., e n+1.setu 1 = A (θ D and set U = A ( θ E. Notice that (U 1 + U can be written as a sum of four orthogonal matrices, so that is a sum of six orthogonal matrices. If k, choose θ so that σ 1 cos θ = k. MakethesamechoicesforD and E as in the case k = 3, and also make the same choices for U 1 and U. Notice now that can be written as a sum of k + orthogonal matrices. We summarize our results. Theorem 15. Let A M n+1 (R be given. Suppose k is an integer such that σ 1 (A k. If k 3, then A can be written as a sum of at most six matrices in O n+1 (R. If k, then A can be written as a sum of k + matrices in O n+1 (R. Let A M n+1 (R be given. Write A = P Q, with the singular values arranged in decreasing order, that is, σ 1 σ n+1. Let U = diag(u 1,..., u n+1, where u i = 1ifσ i 1 and u i = 1 otherwise. Consider U. Then, we have subtracted 1 to those singular values that are bigger than

8 D.I. Merino / Linear Algebra and its Applications 36 ( and we have added 1 to the singular values that are less than 1. Suppose that σ 1 k and k >. Repeating this process k 3 more times results in a matrix whose (diagonal entries are only between 0 and. Suppose that σ 1. Then B diag(σ 1,σ can be written as a sum of four or more orthogonal matrices, say B = t i=1 Q i. If t is even, set P i = Q i [ ( 1 i+1]. Let C = diag(σ 1,σ,σ 3 and let D = t i=1 P i. Then C D = diag(0, 0,σ 3 can be written as a sum of four orthogonal matrices. If t is odd and if σ 3 1, set P i = Q i [ ( 1 i+1]. Then C D = diag(0, 0,σ 3 1 is a sum of four orthogonal matrices. If t is odd and if σ 3 < 1, set P i = Q i [ ( 1 i].thenc D = diag(0, 0,σ is a sum of four orthogonal matrices. Hence, A can be written as a sum of k + 6 or more orthogonal matrices..3. The case F = H Let H be the set of quaternions with real coefficients, that is, let H = {a + bi + cj +dk : a, b, c, d R}. Let M n (H be the set of n-by-n matrices with entries from H. See [6] for a discussion on M n (H. Notice that U 1 (H = {z H : z = 1}. Letz = a + bi + cj + dk H be given. Then z = x + yj, where x = a + bi C and y = c + di C. Now, x and y can be written as a sum of elements of U 1 (C. Note that if p U 1 (C,thenp and pj are elements of U 1 (H. One checks that U n (H forms a group under multiplication, so that Lemma 1 applies. Let n. Then, xi and yi M n (C can be written as a sum of matrices from U n (C. Now, if U U n (C, then U and Uj U n (H. Hence, zi can be written as a sum of matrices in U n (H. Following the proof of Lemma, this shows that αe 11 can be written as a sum of matrices from U n (H. Now,forpermutation matrices P, Q U n (H, noticethatp (αe 11 Q = αpe 11 Q,asα H commutes with real matrices. Thus, every A M n (H can be written as a sum of matrices from U n (H. Let A M n (H be given, and write A = U V. Suppose that σ 1 (A k, and suppose that k. Then canbewrittenasasumofk unitary matrices in M n (C. Hence, A canbewrittenasasumof k matrices from U n (H. Theorem 16. Let n be a given positive integer. Let A M n (H be given. Then A can be written as a sum of matrices from U n (H. Moreover,ifk is an integer such that σ 1 (A k, then A can be written as a sum of k matrices from U n (H. Let A, B M n (H be given. We say that A is orthogonal if AA T = I. Notice that (AB T is not necessarily equal to B T A T.Whenn = 1, the equality fails to hold because multiplication of quaternions i j is not commutative. Let C = i and let D = j. One checks that both C and D are orthogonal. However, CD = + k j i is not orthogonal, that is, the set of i j + k orthogonal matrices in M (H does not form a group under multiplication. Let n. Set E = C I n and set F = D I n. Then, E, F O n (H but EF / O n (H. Hence, O n (H does not form a group under multiplication. Let an integer n and let X, Y M n (R. Then X + Yi M n (C can be written as a sum of matrices from O n (C. Similarly, X + Yj can be written as a sum of matrices from O n (R + Rj and X + Yk can be written as a sum of matrices from O n (R + Rk. Hence, every A M n (H can be written as a sum of matrices from O n (H. Theorem 17. Let n be a given integer. Let A M n (H be given. Then A can be written as a sum of matrices from O n (H.

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