# Lecture 1: Schur s Unitary Triangularization Theorem

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1 Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections 21, 22, and 24 of the textbook 1 Unitary matrices Definition 1 A matrix U M n is called unitary if UU I ( U U) If U is a real matrix (in which case U is just U ), then U is called an orthogonal matrix Observation: If U, V M n are unitary, then so are Ū, U, U ( U 1 ), UV Observation: If U is a unitary matrix, then det U 1 Examples: Matrices of reflection and of rotations are unitary (in fact, orthogonal) matrices For instance, in 3D-space, 1 reflection along the z-axis: U 1, det U 1, 1 cos θ sin θ rotation along the z-axis: U sin θ cos θ, det U 1 1 That these matrices are unitary is best seen using one of the alternate characterizations listed below Theorem 2 Given U M n, the following statements are equivalent: (i) U is unitary, (ii) U preserves the Hermitian norm, ie, Ux x for all x C n (iii) the columns U 1,, U n of U form an orthonormal system, ie, { U i, U j δ i,j, in other words Uj 1 if i j, U i if i j 1

2 Proof (i) (ii) If U U I, then, for any x C n, Ux 2 2 Ux, Ux U Ux, x x, x x 2 2 (ii) (iii) Let (e 1, e 2,, e n ) denote the canonical basis of C n Assume that U preserves the Hermitian norm of every vector We obtain, for j {1,, n}, U j, U j U j 2 2 Ue j 2 2 e j Moreover, for i, j {1,, n} with i j, we have, for any complex number λ of modulus 1, R λu i, U j 1 ( λui + U j 2 2 U i 2 2 U j 2 1 ( 2 2) U(λei + e j ) 2 2 U(e i ) 2 2 U(e j ) 2 2 2) 1 ( λei + e j 2 2 e i 2 2 e j 2 ) 2 2 This does imply that U i, U j (argue for instance that λu i, U j R λu i, U j for a properly chosen λ C with λ 1) (iii) (i) Observe that the (i, j)th entry of U U is U i U j to realize that (iii) directly translates into U U I According to (iii), a unitary matrix can be interpreted as the matrix of an orthonormal basis in another orthonormal basis In terms of linear maps represented by matrices A, the change of orthonormal bases therefore corresponds to the transformation A UAU for some unitary matrix U This transformation defines the unitary equivalence Definition 3 Two matrices A, B M n are called unitary equivalent if there exists a unitary matrix U M n such that B UAU Observation: If A, B M n are unitary equivalent, then a i,j 2 b i,j 2 Indeed, recall form Lect1-Ex3 that a i,j 2 tr (A A) and b i,j 2 tr (B B), and then write tr ( B B ) tr ( (UAU ) (UAU ) ) tr ( UA U UAU ) tr ( UA AU ) tr ( U UA A ) tr ( A A ) 2

3 2 Statement of Schur s theorem and some of its consequences Schur s unitary triangularization theorem says that every matrix is unitarily equivalent to a triangular matrix Precisely, it reads as follows Theorem 4 Given A M n with eigenvalues λ 1,, λ n, counting multiplicities, there exists a unitary matrix U M n such that λ 1 x x A U λ 2 x U λ n Note that such a decomposition is far from unique (see Example 232 p8-81) Let us now state a few consequences from Schur s theorem First, Cayley Hamilton theorem says that every square matrix annihilates its own characteristic polynomial Theorem 5 Given A M n, one has p A (A) The second consequence of Schur s theorem says that every matrix is similar to a blockdiagonal matrix where each block is upper triangular and has a constant diagonal This is an important step in a possible proof of Jordan canonical form Theorem 6 Given A M n with distinct eigenvalues λ 1,, λ k, there is an invertible matrix S M n such that T 1 λ i x x A S T 2 S 1, where T i has the form T i λ i x T k λ i The arguments for Theorems 5 and 6 (see next section) do not use the unitary equivalence stated in Schur s theorem, but merely the equivalence The unitary equivalence is nonetheless crucial for the final consequence of Schur s theorem, which says that there are diagonalizable matrices arbitrary close to any matrix (in other words, the set of diagonalizable matrices is dense in M n ) Theorem 7 Given A M n and ε >, there exists a diagonalizable matrix Ã M n such that a i,j ã i,j 2 < ε 3

4 3 Proofs Proof of Theorem 4 We proceed by induction on n 1 For n 1, there is nothing to do Suppose now the result true up to an integer n 1, n 2 Let A M n with eigenvalues λ 1,, λ n, counting multiplicities Consider an eigenvector v 1 associated to the eigenvalue λ 1 We may assume that v We use it to form an orthonormal basis (v 1, v 2,, v n ) The matrix A is equivalent to the matrix of the linear map x Ax relative to the basis (v 1, v 2,, v n ), ie, λ 1 x x (1) A V Ã V 1, ] where V [v 1 v n is the matrix of the system (v 1, v 2,, v n ) relative to the canonical basis Since this is a unitary matrix, the equivalence of (1) is in fact a unitary equivalence Note that p A (x) (λ 1 x)pã(x), so that the eigenvalues of Ã M n 1, counting multiplicities, are λ 2,, λ n We use the induction hypothesis to find a unitary matrix W M n 1 such that λ 2 x x λ 2 x x Ã W W, ie, W Ã W x x λ n λ n Now observe that 1 λ 1 x x 1 1 λ 1 x x W Ã W W Ã W λ 1 x x λ 1 x x W Ã W λ 2 x λ n 1 Since W : is a unitary matrix, this reads W λ 1 x x λ 1 x x (2) Ã W λ 2 x W λ n Putting the unitary equivalences (1) and (2) together shows the result of Theorem 4 (with U V W ) for the integer n This concludes the inductive proof 4

5 Now that Schur s theorem is established, we may prove the consequences stated in Section 2 Proof of Theorem 5 First attempt, valid when A is diagonalizable In this case, there is a basis (v 1,, v n ) of eigenvectors associated to (not necessarily distinct) eigenvalues λ 1,, λ n It is enough to show that the matrix p A (A) vanishes on each basis vector v i, 1 i n Note that p A (A) (λ 1 I A) (λ n I A) [(λ 1 I A) (λ i 1 I A)(λ i+1 I A) (λ n I A)](λ i I A), because (λ i I A) commutes with all (λ j I A) Then the expected results follows from p A (A)(v i ) [ ](λ i I A)(v i ) [ ]() Final proof Let λ 1,, λ n be the eigenvalues of A M n, counting multiplicities According to Schur s theorem, we can write λ 1 x x A ST S 1 λ, where T 2 x λ n Since P (A) SP (T )S 1 for any polynomial P, we have in particular p A (A) Sp A (T )S 1, so it is enough to show that p A (T ) We compute p A (T ) (λ 1 I T )(λ 2 I T ) (λ n I T ) x x x x x x x x x x x x x x X x x X X x x Using repeatedly the observation about multiplication of block-triangular matrices that x x x x x x x x x x, x x x the zero-block on the top left propagates until we obtain p A (T ) a rigorous proof of this propagation statement should proceed by induction To establish the next consequence of Schur s theorem, we will use the following result Lemma 8 If A M m and B M n are two matrices with no eigenvalue in common, then the matrices A M A and B B are equivalent for any choice of M M m n 5

6 Proof For X M m n, consider the matrices S and S 1 given by I X S and S 1 I X I I We compute S 1 A S B [ A ] AX XB B The result will follow as soon as we can find X M m n such that AX XB M for our given M M m n If we denote by F the linear map F : X M m n AX XB M m n, it is enough to show that F is surjective But since F is a linear map from M m n into itself, it is therefore enough to show that F is injective, ie, that (3) AX XB? X To see why this is true, let us consider X M m n such that AX XB, and observe that A 2 X A(AX) A(XB) (AX)B (XB)B XB 2, A 3 X A(A 2 X) A(XB 2 ) (AX)B 2 (XB)B 2 XB 3, etc, so that P (A)X XP (B) for any polynomial P If we choose P p A as the characteristic polynomial of A, Cayley Hamilton theorem implies (4) Xp A (B) Denoting by λ 1,, λ n the eigenvalues of A, we have p A (B) (λ 1 I B) (λ n I B) Note that each factor (λ i I B) is invertible, since none of the λ i is an eigenvalue of B, so that p A (B) is itself invertible We can now conclude from (4) that X This establishes (3), and finishes the proof We could have given a less conceptual proof of Lemma 8 in case both A and B are upper triangular (see exercises), which is actually what the proof presented below requires Proof of Theorem 6 For A M n, we sort its eigenvalues as λ 1,, λ 1 ;λ 2,, λ 2 ; ;λ k,, λ k, counting multiplicities Schur s theorem guarantees that A is similar to the matrix T 1 X X λ i x x T 2 where T i λ i X x T k λ i 6

7 We now use the Lemma 8 repeatedly in the chain of equivalences T 1 X X T 1 T 1 T 2 X X T 2 X X T 2 X A T 3 T 3 T 3 X X X T k T k T k T 1 T 1 T 1 T 2 T 2 X T 2 T 3 X T 3 T 3 X T k T k T k T 1 T 2 T k This is the announced result Proof of Theorem 7 Let us sort the eigenvalues of A as λ 1 λ n According to Schur s theorem, there exists a unitary matrix U M n such that λ 1 x x A U λ 2 x U λ n If λ i : λ i + iη and η > is small enough to guarantee that λ 1,, λ n are all distinct, we set λ 1 x x Ã U λ2 x U λn In this case, the eigenvalues of Ã (ie, λ 1,, λ n ) are all distinct, hence Ã is diagonalizable We now notice that ) a i,j ã i,j 2 tr ((A Ã) (A Ã) But since A Ã is unitarily equivalent of the diagonal matrix diag[λ 1 λ 1,, λ n λ n ], this quantity equals 1 i n λ i λ i 2 It follows that a i,j ã i,j 2 i 2 η 2 < ε, 1 i n provided η is chosen small enough to have η 2 < ε/ i i2 7

8 4 Exercises Ex1: Is the sum of unitary matrices also unitary? Ex2: Exercise 2 p 71 Ex3: When is a diagonal matrix unitary? Ex4: Exercise 12 p 72 Ex5: Given A M n with eigenvalues λ 1,, λ n, counting multiplicities, prove that there exists a unitary matrix U M n such that λ 1 A U x λ 2 U x x λ n Ex6: Prove that a matrix U M n is unitary iff it preserves the Hermitian inner product, ie, iff Ux, Uy x, y for all x, y C n Ex7: Given a matrix A M n with eigenvalues λ 1,, λ n, counting multiplicities, and given a polynomial P (x) c d x d + + c 1 x + c, prove that the eigenvalues of P (A) are P (λ 1 ),, P (λ n ), counting multiplicities (Note: this is not quite the same as Exercise 7 from Lecture 1) Ex8: Exercise 5 p 97 Ex9: For any matrix A M n, prove that det(exp(a)) exp(tr (A)) (The exponential of a matrix is defined as the convergent series exp(a) k 1 k! Ak ) Ex1: Given an invertible matrix A M n, show that its inverse A 1 can be expressed as a polynomial of degree n 1 in A Ex11: Without using Cayley-Hamilton theorem, prove that if T M m and T M n are two upper triangular matrices with no eigenvalue in common, then the matrices T M T and T T are equivalent for any choice of M M m n [Hint: Observe that you need to show T X X T X Start by considering the element in the lower left corner of the matrix T X X T to show that x m,1, then consider the diagonal i j m 2 (the one just above the lower left corner) of the matrix T X X T to show that x m 1,1 and x m,2, etc] 8

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