Principles of Mathematics in Operations Research

64 4 Eigen Values and Vectors

Im

a+ib= r (coxO + i sinO)

*- Re

a

a+ib=a-ib

Fig. 4.2. Complex conjugate

Definition 4.5.2 A = AH with entries (AH)ij = (A)^ is known as conju-

gate transpose (Hermitian transpose).

Properties:

i. < x, y >— xHy, x A. y <=> xHy = 0,

ii. \\x\\ = (xHx)z,

in. {AB)H = BHAH.

Definition 4.5.3 A is Hermitian if AH = A.

Properties:

i. A" = A,~ix<E C", xHAx € K".

ii. Every eigen value of a Hermitian matrix is real.

Hi. The eigen vectors of a Hermitian matrix, if they correspond to different

eigen values, are orthogonal to each other,

iv. (Spectral Theorem)

A = AH, there exists a diagonalizing unitary (complex matrix of orthonor-

mal vectors as columns) U such that

U'^1 AU = UHAU = A.

Therefore, any Hermitian matrix can be decomposed into

A = USUH = Xivxv? + ••• + \„vnv%.

Definition 4.5.4 If B = M~^1 AM (change of variables), then A and B have
the same eigen values with the same multiplicities, termed as A is similar to
B.