MATRIX THEORY CHAPTER 2 FALL 2017 1.UNITARY MATRIX Definition 1.A matriz UE Mn(C)is said to be unitary if UU=I. Theorem 1.IfUEM,the following are equivalent. (c)UUU (d)The column vectors of U form an orthonomal basis of C". Remark 1.All nonsingular matrices form a group GL(nC):all unitary matrices form a group U(n).which is a subgroup of GL(n,C). Example 1.U= Uesa nlation on the pne. wEC.V defines a refection w.r.t.the plane Example 3.U= 2.UNITARY EQUIVALENCE iroe tr ent teearr ck Theorem 2.Unitary equivalence is an equivalent relation.ie.if denoting "B is unitarily equivalent to A" bu B~A.then ·refective:A~A: symmetric:B~A implies A~B; .transitive:CB and B~A imply C~A. Theorem 3.If A=and Bare unitary equivalent,then ∑∑1aP=∑bP Remark 2.Unitary equivalence is finer than similarity. Example 4.A 32+12+(2)2+21++02+22,they are not unitarily equivalent
MATRIX THEORY - CHAPTER 2 FALL 2017 1. Unitary Matrix Definition 1. A matrix U ∈ Mn(C) is said to be unitary if U ∗U = I. Theorem 1. If U ∈ Mn, the following are equivalent: (a) U ∗U = I, i.e. U is unitary. (b) U is nonsingular and U −1 = U ∗ . (c) UU∗ = I, i.e. U ∗ is unitary. (d) The column vectors of U form an orthonomal basis of C n. (e) The row vectors of U form an orthonomal basis of C n. (f) The map x → Ux is an isometry, i.e. for all x ∈ C n, kxk = kUxk. Remark 1. All nonsingular matrices form a group GL(n, C); all unitary matrices form a group U(n), which is a subgroup of GL(n, C). All nonsingular real matrices form a group GL(n, R); all unitary real matrices (also called real orthogonal matrices) form a group O(n), which is a subgroup of GL(n, R). Example 1. U = cos θ sin θ − sin θ cos θ : U defines a rotation on the plane. Example 2. Uw = I − 2(w ∗w) −1ww∗ , where 0 6= w ∈ C n. Uw defines a reflection w.r.t. the plane perpendicular to w, which is called the Householder transformation. Example 3. U = √ 1 2 √−1 3 √ 1 6 0 √ 1 3 √ 2 6 √ 1 2 √ 1 3 √−1 6 2. Unitary equivalence Definition 2. A matrix B is said to be unitarily equivalent to A if there exists some unitary matrix U such that B = U ∗AU. Theorem 2. Unitary equivalence is an equivalent relation, i.e. if denoting ”B is unitarily equivalent to A” by B ∼ A, then • reflective: A ∼ A; • symmetric: B ∼ A implies A ∼ B; • transitive: C ∼ B and B ∼ A imply C ∼ A. Theorem 3. If A = [aij ] and B = [bij ] are unitary equivalent, then X ij |aij | 2 = X ij |bij | 2 . Remark 2. Unitary equivalence is finer than similarity. Example 4. A = 3 1 −2 0 , B = 1 1 0 2 . Since σ(A) = σ(B) = {1, 2}, there are similar. Since 3 2 + 12 + (−2)2 + 02 6= 12 + 12 + 02 + 22 , they are not unitarily equivalent. 1
2 FALL 2017 3.SCHUR'S LEMMA ).C),tere r U'AU=T=tul is upper triangular with diagonal entries. Proof.Induction method.When n=1,it is obviously true.Assume the theorem is true for (n-1)x(n-1) matrices,we prove it is also true n x n matrices. Remark 3.Notice that of A.We can arrange them in any order by choosing 4.IMPLICATIONS OF SCHUR'S THEOREM Theorem 5 (Cayley-Hamilton).For any A Mn(C),let pa(t)=det(tI-A).Then pA(A)=0. is done by dire A( Corollary 1.If AE M(C)is nonsingular,then there erists a polynomial q(t)of order <n-1 such that A-1=g(A). A2=34-21,43=A(34-2)=7A-61,… and A1-24-30,42-4-3P=-A+1 … -Ar…-Arr如 「 T= T= T Proof.First,we can choose UE U(n)such that UAU=T,where T=j]is upper triangular such that ,2...Consider tra with r<s and (I+aE)T(I-aE) all such t in the same way as above in the order (n (n2n1n2.).(nn2).(n t..becomes t. .+at..-at...Hence by choosing 3,n-1),(n -3,n),....This can make all (r,s)-entry to be zero if r<s,trt. Remark 4.The elementary ro/column operations can be represented by matrir product from the left/right. multiply the r-th row o时Aba卡0：（I+(a-1)Er)A multiply the r-th column of A by 0:A(I+(a-1)Err)
2 FALL 2017 3. Schur’s Lemma Theorem 4 (Schur). Given A ∈ Mn(C) with eigenvalues λ1, ..., λn in any prescribed order, there is a unitary matrix U ∈ Mn(C) such that U ∗AU = T = [tij ] is upper triangular with diagonal entries λ1, ..., λn. Proof. Induction method. When n = 1, it is obviously true. Assume the theorem is true for (n−1)×(n−1) matrices, we prove it is also true n × n matrices. Remark 3. Notice that • In Schur’s theorem, neither T nor U is unique. • The diagonal entries of T are the eigenvalues of A. We can arrange them in any order by choosing suitable U. 4. Implications of Schur’s theorem Theorem 5 (Cayley-Hamilton). For any A ∈ Mn(C), let pA(t) = det(tI − A). Then pA(A) = 0. Proof. By Schur’s theorem, there exists some unitary matrix U and upper triangular matrix T such that U ∗AU = T. Then pA(A) = U ∗pA(T)U. So we only need to show pA(T) = (T − λ1I)· · ·(T − λnI) = 0. This is done by direct computation. Corollary 1. If A ∈ Mn(C) is nonsingular, then there exists a polynomial q(t) of order ≤ n − 1 such that A−1 = q(A). Example 5. A = 3 1 −2 0 , pA(t) = t 2 − 3t + 2. Hence A2 − 3A + 2I = 0. This implies A 2 = 3A − 2I, A3 = A(3A − 2I) = 7A − 6I, · · · and A −1 = −1 2 (A − 3I), A−2 = 1 4 (A − 3I) 2 = − 13 8 A + 21 8 I, · · · Theorem 6 (Block diagonalization). For A ∈ Mn(C), let pA(t) = (t − λ1) n1 · · ·(t − λk) nk where λ1, ..., λk are distinct eigenvalues. Then A is similar to a block diagonal matrix T = T1 T2 . . . Tk , Ti = λi ∗ ∗ . . . ∗ λi . Proof. First, we can choose U ∈ U(n) such that U ∗AU = T, where T = [tij ] is upper triangular such that the diagonal entries are arranged as λ1, ..., λ1, λ2, ....λ2, ..., λk, ..., λk. Consider trs with r < s and trr 6= tss. Then after the following similar transformation, (I + αErs)T(I − αErs) trs becomes trs + αtss − αtrr. Hence by choosing α = trs trr−tss , the (r, s)-entry is zero. Finally, we deal with all such trs in the same way as above in the order (n − 1, n),(n − 2, n − 1),(n − 2, n),(n − 3, n − 2),(n − 3, n − 1),(n − 3, n), · · · . This can make all (r, s)-entry to be zero if r < s, trr 6= tss. Remark 4. The elementary row/column operations can be represented by matrix product from the left/right. • Type I. exchange r-th row and s-th row of A (s 6= r): (I − Err − Ess + Ers + Esr)A; exchange r-th column and s-th column of A (s 6= r): A(I − Err − Ess + Ers + Esr). • Type II. multiply the r-th row of A by α 6= 0: (I + (α − 1)Err)A; multiply the r-th column of A by α 6= 0: A(I + (α − 1)Err).
MATRIX THEORY CHAPTER 2 3 ·Tpe the s-th ro n of a b o and add it to the r-th (I+aE)A multiply the r-th column of A by and add it to the s-th row:A(): Example 6. T- 5 We will make similr trumsformation tsuch that the small bockbecomes sero.We ill deal with the entries in the order(3,4),(2,3),(2,4),(1,2),(1,3),(L,4). .(3,4):since tgs=t=5,we can not make this entry to be zero and just leave it there. 。（2,4):since t22丰t4,ve can m e a similar transformation 213 4-a1 「21361 T'-(I+aE24)T(I-E)= 206+5a1-2a1 200 0 5 (a--2) (1,2):since=2,we can not make this entry to be zero and just leave it there. 。（,3):since t1≠tg,ve can make a similar tran.sformation 「213+5a2-2a26+7a21「210-1 T"=(I+a2E13)T'(I-a2E1g)= 2 0 0 200 5 (a2=-1) .(1,4):since we can make a similar transformation 210-1+5a-2ag 21001 T"M=(I+a3E14)T"(I-a3E1)= 20 200 5 7 52 (ag=) 5 5 Denote S=(I+a3E)(I+2E3)(I+E2).Then STS-1=T"is a block diagonal matrir 5.NORMAL MATRIX Definition 3.A matrir AE M(C)is said to be normal if A'A=AA" Example 7.The following are normal matrices: .Unitaru matrir:U'U =I 。Hermitian matrir:.A'=A .Skew-Hermitian matrir:A*=-A Theorem 7.If A is normal and B=UAU for some unitary matrir U,then B is also normal Theorem 8.For AEM(C),the following are equivalent mal (∑ thA-人re te时A (d)There is an orthonormal set ofn eigenvectors of A. Theorem 9.If AE M(C)is Hermitian,then
MATRIX THEORY - CHAPTER 2 3 • Type III. multiply the s-th row of A by α and add it to the r-th row: (I + αErs)A; multiply the r-th column of A by α and add it to the s-th row: A(I + αErs); Example 6. T = 2 1 3 4 2 0 6 5 7 5 We will make similar transformation to T such that the small block 3 4 0 6 becomes zero. We will deal with the entries in the order (3, 4),(2, 3),(2, 4),(1, 2),(1, 3),(1, 4). • (3, 4): since t33 = t44 = 5, we can not make this entry to be zero and just leave it there. • (2, 3): since t23 is already zero, we don’t need to make any change. • (2, 4): since t22 6= t44, we can make a similar transformation T 0 = (I + α1E24)T(I − α1E24) = 2 1 3 4 − α1 2 0 6 + 5α1 − 2α1 5 7 5 = 2 1 3 6 2 0 0 5 7 5 (α1 = −2) • (1, 2): since t 0 11 = t 0 22 = 2, we can not make this entry to be zero and just leave it there. • (1, 3): since t 0 11 6= t 0 33, we can make a similar transformation T 00 = (I + α2E13)T 0 (I − α2E13) = 2 1 3 + 5α2 − 2α2 6 + 7α2 2 0 0 5 7 5 = 2 1 0 −1 2 0 0 5 2 5 (α2 = −1) • (1, 4): since t 0 11 6= t 0 44, we can make a similar transformation T 000 = (I + α3E14)T 00(I − α3E14) = 2 1 0 −1 + 5α3 − 2α3 2 0 0 5 7 5 = 2 1 0 0 2 0 0 5 2 5 (α3 = 1 3 ) Denote S = (I + α3E14)(I + α2E13)(I + α1E24). Then ST S−1 = T 000 is a block diagonal matrix. 5. Normal matrix Definition 3. A matrix A ∈ Mn(C) is said to be normal if A∗A = AA∗ . Example 7. The following are normal matrices: • Unitary matrix: U ∗U = I • Hermitian matrix: A∗ = A • Skew-Hermitian matrix: A∗ = −A Theorem 7. If A is normal and B = U ∗AU for some unitary matrix U, then B is also normal. Theorem 8. For A ∈ Mn(C), the following are equivalent (a) A is normal. (b) A is unitary diagonalisable. (c) Pn i,j=1 |aij | 2 = Pn i |λi | 2 , where λ1, ..., λn are the eigenvalues of A. (d) There is an orthonormal set of n eigenvectors of A. Theorem 9. If A ∈ Mn(C) is Hermitian, then (a) all eigenvalues of A is real. (b) A is unitary diagonalisable
4 FALL 2017 If AM(R)is symmetric,then A is real orthogonally diagonalisable. det(tI-A)=(t-1j(t+1). -1a-方(）a-a =-=(）a= Here form an orthonormal basis of C2.Define Then U is unitary and ra=[1-] Example 9.A= A is real symmetric and hence normal.The characteristic polynomial -148 of A is det(tI-A)=(t-9)2(t+9). =9,= 0,Aa2=22 1 Here a,02,as form orthonormal basis of R3.Define Q=[a1,02.03]= Then Q is real orthogonal and 「9 9 -9
4 FALL 2017 If A ∈ Mn(R) is symmetric, then A is real orthogonally diagonalisable. Example 8. A = 0 i −i 0 . A is Hermitian and hence normal. The characteristic polynomial of A is det(tI − A) = (t − 1)(t + 1). λ1 = 1, α1 = 1 √ 2 i 1 , Aα1 = λ1α1, λ2 = −1, α2 = 1 √ 2 1 −i , Aα2 = λ2α2. Here α1, α2 form an orthonormal basis of C 2 . Define U = [α1, α2] = " √ i 2 √ 1 2 √ 1 2 √−i 2 # . Then U is unitary and U ∗AU = 1 −1 . Example 9. A = 8 4 −1 4 −7 4 −1 4 8 . A is real symmetric and hence normal. The characteristic polynomial of A is det(tI − A) = (t − 9)2 (t + 9). λ1 = 9, α1 = 1 3 2 1 2 , Aα1 = λ1α1, λ2 = 9, α2 = 1 √ 2 1 0 −1 , Aα2 = λ2α2, λ3 = −9, α3 = 1 3 √ 2 1 −4 1 , Aα3 = λ3α3, Here α1, α2, α3 form orthonormal basis of R 3 . Define Q = [α1, α2, α3] = 2 3 √ 1 2 1 3 √ 2 1 3 0 −4 3 √ 2 2 3 √−1 2 1 3 √ 2 . Then Q is real orthogonal and Q T AQ = 9 9 −9
