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 2.1, 2.2, and 2.4 of the textbook.
1 Unitary matrices
Definition 1. A matrix U ∈ Mn 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.
> ∗ −1 Observation: If U, V ∈ Mn are unitary, then so are U¯, U , U (= U ), 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 0 0 reflection along the z-axis: U = 0 1 0 , det U = −1, 0 0 −1 cos θ − sin θ 0 rotation along the -axis: z U = sin θ cos θ 0 , det U = 1. 0 0 1
That these matrices are unitary is best seen using one of the alternate characterizations listed below.
Theorem 2. Given U ∈ Mn, the following statements are equivalent: (i) U is unitary,
(ii) U preserves the Hermitian norm, i.e.,
n kUxk = kxk for all x ∈ C .
(iii) the columns U1,...,Un of U form an orthonormal system, i.e., ( 1 if i = j, hU ,U i = δ , in other words U ∗U = i j i,j j i 0 if i 6= j.
1 ∗ n Proof. (i) ⇒ (ii). If U U = I, then, for any x ∈ C ,
2 ∗ 2 kUxk2 = hUx, Uxi = hU Ux, xi = hx, xi = kxk2.
n (ii) ⇒ (iii). Let (e1, e2, . . . , en) denote the canonical basis of C . Assume that U preserves the Hermitian norm of every vector. We obtain, for j ∈ {1, . . . , n},
2 2 2 hUj,Uji = kUjk2 = kUejk2 = kejk2 = 1.
Moreover, for i, j ∈ {1, . . . , n} with i 6= j, we have, for any complex number λ of modulus 1, 1 1