# 5. Orthogonal matrices

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1 L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1

2 Orthonormal vectors a set of real m-vectors {a 1, a 2,, a n } is orthonormal if the vectors have unit norm: a i = 1 they are mutually orthogonal: a T i a j = 0 if i j Example 0 0 1, , Orthogonal matrices 5-2

3 Matrix with orthonormal columns A R m n has orthonormal columns if its Gram matrix is the identity A T A = [ a 1 a 2 a n ] T [ a1 a 2 a n ] = = a T 1 a 1 a T 1 a 2 a T 1 a n a T 2 a 1 a T 2 a 2 a T 2 a n a T na 1 a T na 2 a T na n there is no standard short name for matrix with orthonormal columns Orthogonal matrices 5-3

4 Matrix-vector product if A R m n has orthonormal columns, then the linear function f(x) = Ax preserves inner products: (Ax) T (Ay) = x T A T Ay = x T y preserves norms: Ax = ( (Ax) T (Ax) ) 1/2 = (x T x) 1/2 = x preserves distances: Ax Ay = x y preserves angles: (Ax, Ay) = arccos ( (Ax) T ) (Ay) Ax Ay = arccos ( x T y ) x y = (x, y) Orthogonal matrices 5-4

5 Left invertibility if A R m n has orthonormal columns, then A is left invertible with left inverse A T : by definition A T A = I A has linearly independent columns (from p 4-24 or p 5-2): Ax = 0 = A T Ax = x = 0 A is tall or square: m n (see page 4-13) Orthogonal matrices 5-5

6 Outline matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns

7 Orthogonal matrix Orthogonal matrix: a square real matrix with orthonormal columns Nonsingularity (from equivalences on page 4-14): if A is orthogonal, then A is invertible, with inverse A T : A T A = I A is square } = AA T = I A T is also an orthogonal matrix rows of A are orthonormal (have norm one and are mutually orthogonal) Note: if A R m n has orthonormal columns and m > n, then AA T I Orthogonal matrices 5-6

8 Permutation matrix let π = (π 1, π 2,, π n ) be a permutation (reordering) of (1, 2,, n) we associate with π the n n permutation matrix A A iπi = 1, A ij = 0 if j π i Ax is a permutation of the elements of x: Ax = (x π1, x π2,, x πn ) A has exactly one element equal to 1 in each row and each column Orthogonality: permutation matrices are orthogonal A T A = I because A has exactly one element equal to one in each row (A T A) ij = n A ki A kj = k=1 { 1 i = j 0 otherwise A T = A 1 is the inverse permutation matrix Orthogonal matrices 5-7

9 Example permutation on {1, 2, 3, 4} (π 1, π 2, π 3, π 4 ) = (2, 4, 1, 3) corresponding permutation matrix and its inverse A = , A 1 = A T = A T is permutation matrix associated with the permutation ( π 1, π 2, π 3, π 4 ) = (3, 1, 4, 2) Orthogonal matrices 5-8

10 Plane rotation Rotation in a plane Ax A = [ cos θ sin θ sin θ cos θ ] θ x Rotation in a coordinate plane in R n : for example, A = cos θ 0 sin θ sin θ 0 cos θ describes a rotation in the (x 1, x 3 ) plane in R 3 Orthogonal matrices 5-9

11 Reflector A = I 2aa T with a a unit-norm vector ( a = 1) a reflector matrix is symmetric and orthogonal A T A = (I 2aa T )(I 2aa T ) = I 4aa T + 4aa T aa T = I Ax is reflection of x through the hyperplane H = {u a T u = 0} H line through a and origin x y z = Ax y = x (a T x)a = (I aa T )x z = y + (y x) = (I 2aa T )x (see page 2-35) Orthogonal matrices 5-10

12 Product of orthogonal matrices if A 1,, A k are orthogonal matrices and of equal size, then the product A = A 1 A 2 A k is orthogonal: A T A = (A 1 A 2 A k ) T (A 1 A 2 A k ) = A T k A T 2 A T 1 A 1 A 2 A k = I Orthogonal matrices 5-11

13 Linear equation with orthogonal matrix linear equation with orthogonal coefficient matrix A of size n n Ax = b solution is x = A 1 b = A T b can be computed in 2n 2 flops by matrix-vector multiplication cost is less than order n 2 if A has special properties; for example, permutation matrix: reflector (given a): plane rotation: 0 flops order n flops order 1 flops Orthogonal matrices 5-12

14 Outline matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns

15 Tall matrix with orthonormal columns suppose A R m n is tall (m > n) and has orthonormal columns A T is a left inverse of A: A T A = I A has no right inverse; in particular AA T I on the next pages, we give a geometric interpretation to the matrix AA T Orthogonal matrices 5-13

16 Range the span of a set of vectors is the set of all their linear combinations: span(a 1, a 2,, a n ) = {x 1 a 1 + x 2 a x n a n x R n } the range of a matrix A R m n is the span of its column vectors: range(a) = {Ax x R n } Example range( ) = x 1 + x 3 x 1 + x 2 + 2x 3 x 2 x 3 x 1, x 2, x 3 R = {(u, u + v, v) u, v R} Orthogonal matrices 5-14

17 Projection on range of matrix with orthonormal columns suppose A R m n has orthonormal columns; we ll show that the vector AA T b is the orthogonal projection of an m-vector b on range(a) b range(a) = {Ax x R n } Aˆx = AA T b ˆx = A T b satisfies Aˆx b < Ax b for all x ˆx this extends the result on page 2-11 (where A = (1/ a )a) Orthogonal matrices 5-15

18 Proof the squared distance of b to an arbitrary point Ax in range(a) is Ax b 2 = A(x ˆx) + Aˆx b 2 (where ˆx = A T b) = A(x ˆx) 2 + Aˆx b 2 + 2(x ˆx) T A T (Aˆx b) = A(x ˆx) 2 + Aˆx b 2 = x ˆx 2 + Aˆx b 2 Aˆx b 2 with equality only if x = ˆx line 3 follows because A T (Aˆx b) = ˆx A T b = 0 line 4 follows from A T A = I Orthogonal matrices 5-16

19 Orthogonal decomposition the vector b is decomposed as a sum b = z + y with z range(a), y range(a) y = b AA T b b range(a) z = AA T b such a decomposition exists and is unique for every b b = Ax + y, A T y = 0 A T b = x, y = b AA T b Orthogonal matrices 5-17

20 Outline matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns

21 Gram matrix A C m n has orthonormal columns if its Gram matrix is the identity: A H A = [ a 1 a 2 a n ] H [ a1 a 2 a n ] = = a H 1 a 1 a H 1 a 2 a H 1 a n a H 2 a 1 a H 2 a 2 a H 2 a n a H n a 1 a H n a 2 a H n a n columns have unit norm: a i 2 = a H i a i = 1 columns are mutually orthogonal: a H i a j = 0 for i j Orthogonal matrices 5-18

22 Unitary matrix Unitary matrix a square complex matrix with orthonormal columns is called unitary Inverse A H A = I A is square } = AA H = I a unitary matrix is nonsingular with inverse A H if A is unitary, then A H is unitary Orthogonal matrices 5-19

23 Discrete Fourier transform matrix recall definition from page 3-33 (with ω = e 2πj/n and j = 1) W = ω 1 ω 2 ω (n 1) 1 ω 2 ω 4 ω 2(n 1) 1 ω (n 1) ω 2(n 1) ω (n 1)(n 1) the matrix (1/ n)w is unitary (proof on next page): inverse of W is W 1 = (1/n)W H 1 n W H W = 1 n W W H = I inverse discrete Fourier transform of n-vector x is W 1 x = (1/n)W H x Orthogonal matrices 5-20

24 we show that W H W = ni Gram matrix of DFT matrix conjugate transpose of W is ω ω 2 ω n 1 W H = 1 ω 2 ω 4 ω 2(n 1) 1 ω n 1 ω 2(n 1) ω (n 1)(n 1) i, j element of Gram matrix is (W H W ) ij = 1 + ω i j + ω 2(i j) + + ω (n 1)(i j) (W H W ) ii = n, (W H W ) ij = ωn(i j) 1 ω i j 1 = 0 if i j (last step follows from ω n = 1) Orthogonal matrices 5-21

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