Chapter 8 - Hermitian and Unitary Matrices PDF

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Chapter 8 Hermitian Matrices and Unitary Matrices 8.1

Hermitian Matrices

Definition 8.1 The Hermitian Transpose of a matrix (denoted by AH ), is the complex conjugate transpose of A, that is AH = (A)T . Definition 8.2 A matrix is normal if AAH = AH A. Theorem 8.3 Every normal matrix is similar to a diagonal matrix. Definition 8.4 A matrix is Hermitian if it equals to it’s own hermitian transpose. i.e. A is hermitian if A = AH . Theorem 8.5 A hermitian matrix is normal. Proof. Let A be a hermitian matrix. Then AAH = AA = AH A. 

Theorem 8.6 Prove that AH A and AAH are normal for any matrix A. Proof. Let A be any matrix. Then • (AH A)H = AH (AH )H = AH A and • (AAH )H = (AH )H AH = AAH . Therefore AAH and AH A are hermitian and consequently normal. 86

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Definition 8.7 Let W be an invertible n × n matrix over C. The inner product of x, y ∈ Cn with respect to W , denoted by hx, yiW is:     hx, yiW = (W x) · W y = (W x)T W y .

where · is the usual dot product and y is the complex conjugate of y .

Note if W = I (the Euclidean inner product), then hx, yi = (x) · ( y ).   1 1 0 Example 8.8 Let x = (−2i, 3− 2i, i) ∈ C3 , y = (2i, 2 + i, 7) ∈ C3 and W = 0 1 1. Calculate 1 0 1 hx, yiW . Solution.   hx, yiW = (W x)T W y    T     −2i 1 1 0 1 1 0 2i   =  0 1 1 3 − 2i    0 1 1 2 + i   1 0 1 i 1 0 1 7     T   −2i + 3 − 2i 2i + 2 + i   =  3 − 2i + i    2 + i + 7   −2i + i 2i + 7   T    3 − 4i 2 + 3i   =  3 − i     9 + i   −i 7 + 2i   2 − 3i   = 3 − 4i 3 − i −i  9 − i  7 − 2i = (3 − 4i)(2 − 3i) + (3 − i)(9 − i) + (−i)(7 − 2i) = 6 − 9i − 8i + 12i2 + 27 − 3i − 9i + i2 − 7i + 2i2 = 18 − 36i. ♣

Theorem 8.9 (Properties of Inner Products) Let x, y, z ∈ Cn and λ ∈ C. Then • hx, xiW is real and positive if x 6= 0. • hx, xiW = 0 iff x = 0. • hx, yiW = hy, xiW .

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• hλx, yiW = λhx, yiW and hx, λyiW = λhx, yiW . • hx + y, ziW = hx, ziW + hy, ziW and hx, y + ziW = hx, yiW + hx, ziW . Definition 8.10 Let W be an invertible n × n matrix and A be an m × n matrix. The adjoint matrix of A (denoted by A∗ ) is an n × m that satisfies: hx, AyiW = hA∗ x, y iW for all x ∈ Cn and y ∈ Cm . Theorem 8.11 The adjoint matrix always exists and is A∗ = (W H W )−1 AH (W H W ). Note for W = I(the Euclidean inner product), A∗ = AH

and

hx, Ayi = hA∗ x, y i = hAH x, yi.

Proof. Let W be an invertible n × n matrix, A is an n × m, A∗ be the adjoint of A and x ∈ Rn or Cn and y ∈ Rm or Cm . Then     • hx, AyiW = (W x) · W Ay = (W x)T W Ay = xT W T W A y.     • hA∗ x, yiW = (W A∗ x) · W y = (W A∗ x)T W y = xT (A∗ )T W T W y.

Now

hx, AyiW = hA∗ x, y iW xT W T W A y = xT (A∗ )T W T W y xT W T W A y − xT (A∗ )T W T W y = 0   xT W T W A − (A∗ )T W T W y = 0   Now xT W T W A − (A∗ )T W T W y = 0 if and only if W T W A − (A∗ )T W T W = 0. Therefore W T W A = (A∗ )T W T W

(A∗ )T W T W = W T W A (A∗ )T = (W T W ) A (W T W )−1  T A∗ = (W T W ) A (W T W )−1  −1  T = (W T W )T A (W T W )T  −1  T = (W )T W A ((W )T W )  H −1 H = W W A (W H W ).

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Example 8.12 Determine the adjoint of A under the inner product with respect to W , where     2 0 i 0 A= and W = . 0 −i 2i −i Solution.  −1 H A (W H W ) A∗ = W H W H  ! H  H  !−1   i 0 i 0 i 0 2 0 i 0 = 0 −i 2i −i 2i −i 2i −i 2i −i       −1  −i −2i i 0 i 0 −i −2i 2 0 = 0 i 2i −i 0 i 0 i 2i −i    2    −1 2 0 −i − 4i2 2i2 −i2 − 4i2 2i2 = 2 2 2i −i 0 i 2i2 −i2 −1     2 0 5 −2 5 −2 = −2 1 0 i −2 1    1 2 10 −4 = −2i i 2 5   10 − 4i −4 + 2i = . 20 − 10i −8 + 5i ♣

Theorem 8.13 Properties of the Adjoint: • (A∗ )∗ = A for any m × n matrix A. • (A + B)∗ = A∗ + B ∗ for any m × n matrices A and B . • (AB)∗ = B ∗ A∗ for any m × n matrices A and B . • (λA)∗ = λA∗ for any m × n matrix A and λ ∈ C. Theorem 8.14 Let A be an n × n matrix. Then the eigenvalues of AH A are nonnegative. Proof. Let A be an n×n matrix and λ be an eigenvalue of AH A. Clearly there exists a corresponding eigenvector v such that AH Av = λv. For the Euclidean inner product, hAv, Avi = hA∗ Av, vi = hAH Av, vi = hλv, vi = λhv, v i.

CHAPTER 8. HERMITIAN MATRICES AND UNITARY MATRICES We know that v 6= 0, therefore hAv, Avi and λhv, vi are nonnegative. Thus λ is nonnegative.

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Theorem 8.15 The eigenvalues of a hermitian matrix are real. Proof. Let A be a hermitian matrix. Let λ be an eigenvalue of A with a corresponding eigenvector v. Then λhv, v i = hλv, v i = hAv, vi = hv, A∗ vi = hv, AH vi = hv, Avi = hv, λvi = λhv, v i. Therefore λ = λ. Clearly λ must be real.

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Theorem 8.16 Let A be a normal matrix and λ be an eigenvalue of A with an associated eigenvector v. Then λ is an eigenvalues of AH with a corresponding eigenvector v . Proof. Let A be a normal matrix and λ be an eigenvalue of A with an associated eigenvector v . Now hAv, Avi = hA∗ Av, vi = hAH Av, vi = hAAH v, vi = hAH v, A∗ vi = hAH v, AH vi. Consider hAv − λv, Av − λvi. Thus hAv − λv, Av − λvi = hAv, Avi − hAv, λv i − hλv, Avi + hλv, λvi = hAH v, AH vi − λ hAv, vi − λhv, Avi + λ λ hv, vi = hAH v, AH vi − λ hv, A∗ vi − λhA∗ v, vi + h λ v, λ v i = hAH v, AH vi − λhAH v, vi − λ hv, AH vi + h λ v, λ vi = hAH v, AH vi − hAH v, λ vi − h λ v, AH vi + h λ v, λ v i = hAH v − λv, AH v − λv i. Clearly Av − λv = 0 and hAv − λv, Av − λvi = 0. Therefore hAH v − λv, AH v − λvi = 0 and AH v − λv = 0. Thus λ is an eigenvalue of AH with a corresponding eigenvector v. 

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Theorem 8.17 Eigenvectors corresponding to distinct eigenvalues of normal matrices are orthogonal with respect to the Euclidean inner product. Proof. Let λ1 and λ2 be two distinct eigenvalues of a normal matrix with corresponding eigenvectors v1 and v2 . Clearly Av1 = λ1 v and Av2 = λ2 v. Now (λ1 − λ2 )hv1 , v2 i = λ1 hv1 , v2 i − λ2 hv1 , v2 i = hλ1 v1 , v2 i − hv1 , λ2 v2 i = hAv1 , v2 i − hv1 , AH v2 i by Theorem 8.16 = hAv1 , v2 i − hv1 , A∗ v2 i = hAv1 , v2 i − hAv1 , v2 i = 0. Clearly λ1 6= λ2 . Therefore hv1 , v2 i = 0 and v1 and v2 are orthogonal.

8.2

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Unitary Matrices

Definition 8.18 A matrix U is unitary if U −1 = U H . Theorem 8.19 A matrix is unitary if and only if it’s rows (or columns) form an orthonormal set of vectors. Proof.(⇒) Let U be a unitary matrix. Designate the rows of U as u1 , u2( , . . . , un . Then the (i, j)th 1 i=j . Therefore element of U U H is hui , uj i. If U is unitary, then U U H = I and hui , uj i = 0 otherwise {u1 , u2 , . . . , un } is an(orthonormal basis. (⇐) Now suppose that the rows of U form an orthonormal 1 i=j . Therefore, U U H = I and U is unitary.  set, then hui , uj i = 0 otherwise Theorem 8.20 The product of unitary matrices of the same type is a unitary matrix. Proof. If A and B are unitary matrices of the same type, then (AB )−1 = B −1 A−1 = B H AH = (AB )H . 

Theorem 8.21 If U is unitary, then hU x, Uy i = hx, y i for all vectors x and y of appropriate dimension.

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Proof. Let U be a unitary matrix and x, y by vectors of appropriate dimension. For the Euclidean inner product: hx, Uyi = hUx, U ∗ Uyi hx, Uyi = hx, U H U yi hx, Uyi = hx, U −1 U yi = hx, yi. 

Theorem 8.22 All eigenvalues of a unitry matrix have absolute values equal to 1. Proof. Let λ ∈ C be an eigenvalue of a unitary matrix U with a corresponding eigenvector v. Then ¯ vi |λ2 |hv, vi = λλhv, = hλv, λvi = hUv, Uvi = hv, vi. 2 Therefore vi = 6 0. If λ = a + ib ∈ C, then λ2 = (a2 − b2 ) + i(2ab) and p |λ | = 1 since hv,p 2 2 2 2 2 2 |λ | = (a − b ) + 4a b = (a + b2 )2 = a2 + b2 . Therefore, if |λ2 | = 1, then |λ| = 1. 

Theorem 8.23 The determinant of a unitary matrix has absolute value equal to 1. Proof. Let U be a unitary matrix, then UU −1 = I det(UU −1 ) = 1 det(UU H ) = 1 det(U )det(U H ) = 1 det(U )det(U T ) = 1 det(U )det(U T ) = 1 det(U )det(U ) = 1. Let z = det(U ) = a + ib ∈ C. If zz = 1, then a2 + b2 = 1 and |det(U )| = 1.

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Theorem 8.24 If U is unitary and A = U H BU , then B is normal if and only if A is normal. Proof. (⇒) Let A = U H BU and suppose that B is normal, then AAH = (U H BU )(U H BU )H = U H B UU H B H U = U H BB H U = U H B H B U = (U H B H )(UU H BU ) = AH A. (⇐) Let B = UAU H and suppose that A is normal, then B H B = (UAU H )H UAU H = UAH U H U AU H = U AH AU H = U AAH U H = (UAU H )(UAH U H ) = BB H ...