Mathematical background Daniele Carnevale Dipartimento di Ing. - - PowerPoint PPT Presentation

Complex numbers Matrices

Mathematical background

Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome “Tor Vergata”

Fondamenti di Automatica e Controlli Automatici

A.A. 2014-2015

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Complex numbers Matrices

The complex number

A complex number z is defined as z = α + jβ ∈ C, α Re{z}, β Im{z}, where α and β are real numbers, j is the imaginary unit such that j2=-1. A different module and phase representation is given by z = ρejθ, ρ = ||z|| = Abs(z), θ = z. = Arg(z) Then ρ is the norm of the “vector” z on the 2D-plane and θ is its angle (counter-clock wise) and ||z||

• Re{z}2 + Im{z}2, z

arccos (Im(z)/||z||) if Im(z) ≥ 0, − arccos (Im(z)/||z||) if Im(z) < 0,

Figure : Different representation of complex numbers.

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Complex numbers Matrices

The complex number cont’d

The sin(·) and cos(·) function can be rewritten using the unitary circle on the complex plane described by z = ejθ with θ ∈ [−π, π] such as sin(θ) = Im(z) ||z|| =

• ||z||=1

Im(z), cos(θ) = Re(z) ||z|| =

• ||z||=1

Re(z).

Figure : Module and phase representation - the unitary circle.

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Complex numbers Matrices

The complex number: algebra

Given two complex numbers z1 = α1 + jβ1, z2 = α2 + jβ2, then z = z1 + z2 = α1 + α2 + j(β1 + β2).

Figure : Addition.

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Complex numbers Matrices

The complex number: subtraction

Given two complex numbers z1 = α1 + jβ1, z2 = α2 + jβ2, then z = z1−z2 = α1−α2 + j(β1−β2).

Figure : Subtraction: exercise....

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Complex numbers Matrices

The complex number: multiplication

Given two complex numbers z1 = α1 + jβ1 = ρ1ejθ1, z2 = α2 + jβ2 = ρ2ejθ2, then z = z1z2 = ρ1ρ2ej(θ1+θ2) ⇒ zm

1 = ρm 1 ejmθ1.

Figure : Multiplication (i → j).

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Complex numbers Matrices

The complex number: division

Given two complex numbers z1 = α1 + jβ1 = ρ1ejθ1, z2 = α2 + jβ2 = ρ2ejθ2, then z = z1 z2 = ρ1 ρ2 ej(θ1−θ2). Another way to do it, define the conjugate of z, z, such as z = Re(z)−jIm(z); z = ρejθ → z = ρe−jθ, zz = ||z||2 = ρ2, then z1 z2 = z1 z2 z2 z2 = z1z2 ||z2||2 = ρ1ejθ1ρ2e−jθ2 ρ2

2

= ρ1 ρ2 ej(θ1−θ2)

Figure : Division: exercise...go on the other way around... .

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Complex numbers Matrices

The complex number: other examples

Further examples: z = ej π

2 = j,

z = e−j π

2 = −j,

z = ejπ = −1, z = e−jπ = −1, z = ejπ/4 = √ 2 2 + j √ 2 2 = cos(π/4) + j sin(π/4),

Figure : Conjugate complex number.

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Complex numbers Matrices

The Euler formula

Given that ejθ = cos(θ) + j sin(θ), then ejθ + e−jθ 2 = cos(θ) + j sin(θ) + cos(−θ) + j sin(−θ) 2 , (1) = cos(θ) + j sin(θ) + cos(θ) − j sin(θ) 2 , (2) = cos(θ). (3) ejθ − e−jθ 2j = cos(θ) + j sin(θ) − (cos(−θ) + j sin(−θ)) 2j , (4) = cos(θ) + j sin(θ) − cos(θ) + j sin(θ) 2j , (5) = sin(θ). (6)

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Complex numbers Matrices

Useful formulas (in the next)

Then, for a complex number z = ρejθ, it holds z + z = ρ

• ejθ + e−jθ

=

• Euler formula

2ρ cos(θ). Given two complex numbers z1 = ρ1ejθ1, z2 = ρ2ejθ2, then z1z2 + z1z2 = ρ1ρ2

• ej(θ1+θ2) + e−j(θ1+θ2)

, = 2ρ1ρ2 cos(θ1 + θ2). (7) Which is the trajectory depicted on the plane C by z(t) = ρ(t)ejθ(t), ρ(t) = exp(−2t), θ(t) = 5t, (8) when t ∈ [0, +∞)? Note that if ρ(t) = 1 and θ(t) = ωt, than cos(ωt) + j sin(ωt) = ejωt.

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Complex numbers Matrices

Roots of complex numbers

The roots of x2 = 1 can be easily found if x ∈ C, e.g x = ±1. The same for x2 = −1, there not exist real x. On the complex plane, the imaginary axis allows to solve also z2 = −1 → z = ±j. Furthermore: z3 = 1: ρ3e3jθ = 1 = ej(2kπ), k ∈ Z ⇐ ⇒ ρ = 1, θ = 2kπ

3 .

(9)

Figure : Roots of z3 = 1.

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Complex numbers Matrices

Roots of complex numbers cont’d

z3 = 1: ρ3e3jθ = 1 = ej(2kπ), k ∈ Z ⇐ ⇒ ρ = 1, θ = 2kπ

3 .

(10) z5 = −1: ρ5e5jθ = −1 = ej(±π+2kπ), k ∈ Z ⇐ ⇒ ρ = 1, θ = ±π+2kπ

5

. (11) zn = −1?

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Complex numbers Matrices

Algebra: Matrices

Consider the matrix Am×n =      a1,1 a1,2 . . . a1,n a2,1 a2,2 . . . a2,n . . . . . . am,1 am,2 . . . am,n      , with m rows and n columns. If n = m the matrix A is said to be “square”, if m > n it is “tall” otherwise “fat”. Based on the coefficients of the matrix A: If ai,j = 0 for all i > j, A is said to be upper triangular If ai,j = 0 for all i < j, A is said to be lower triangular If ai,j = 0 for all i = j, A is said to be diagonal. A square matrix A is symmetric if the transpose matrix A′ is such that A′ = A (see also block diagonal matrices). A is antisymmetric if A′ = −A. The transposed matrix B = A′ is such that bi,j = aj,i. The identity matrix In is a square diagonal matrix n × n with elements on the diagonal equal to 1.

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Complex numbers Matrices

Algebra: Matrices operations

When A and B are matrices with the same number of rows and columns, then A + B = B + A =      . . . . . . . . . . . . ai,j + bi,j . . . . . . . . . . . .      . If the number of columns A is the same number of rows of B, than A × B = C (= B × A not even possibile in some cases), (12) ci,j = ai,1 ai,2 . . . ai,n

• 

    b1,j b2,j . . . bn,j      = [A]i[B]j, (13) where [A]i is the i − th row of the matrix A and [B]j is the j − th column of the matrix B. I × A = A = A × I.

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Complex numbers Matrices

Matrix determinant

The determinant | · | of a matrix is defined as follows: |A| =

n

• j=1

ai,jAi,j =

n

• i=1

ai,jAi,j, (14) where Ai,j is the minor of the element ai,j of the matrix A given by Ai,j = (−1)i+j|Mi,j(A)|, (15) and Mi, j(A) is the sub-matrix obtained eliminating from A the i − th row and the j − th column. This definition is recursive and is based on the fact that if n = 2 then |A| =

• a1,1

a1,2 a2,1 a2,2

• = a1,1a2,2 − a1,2a2,1.

(16)

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Complex numbers Matrices

Matrix determinant cont’d

Example: |A| =

• 

 a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 a3,2 a3,3  

• ,

= a3,1(a1,2a2,3 − a2,2a1,3) − a3,2(a1,1a2,3 − a2,1a1,3) + a3,3(a1,1a2,2 − a1,2a2,1). Other properties: |A| = |A′| B is obtained by exchanging the position of two rows (columns) of A → |B| = −|A| B is obtained by multiplying by λ ∈ R a row (column) of A → |B| = λ|A| B is obtained adding a row (column) of A with another row (column) of A multiplied by a constant → |B| = |A| C = A × B → |C| = |A||B|.

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Complex numbers Matrices

Algebra: inverse matrix

A square matrix is said to be non-singular iff its determinant is different from zero. A is non-singular iff its row vectors [A]i ([Aj]) are independents, i.e. there not exist real coefficients ci such that Ak =

n

• i=1,i=k

ciAi, which means that Ak, for any k = 1..n, can not be defined as a linear combination of the other (subset) rows of A. The same needs to hold for the columns. For square matrix the rank of A, rank(A), is the number of rows (columns) linearly independent, moreover the matrix A is non-singular iff rank(A) = n iff |A| = 0. For non-square matrix Am,n it holds rank(A) ≤ min{m, n}. The null space of A, ker(A), i.e. the subspace of vectors x such that Ax = 0, is such that dim(ker(A)) = n − rank(A).

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Complex numbers Matrices

Algebra: inverse matrix cont’d

For a square non-singular matrix A, the inverse matrix A−1 is such that A × A−1 = I = A−1 × A. The inverse matrix can be defined as A−1 adj(A) |A| , (17) where adj(A)    A1,1 . . . An,1 . . . . . . A1,n . . . An,n    , A adj(A) = adj(A)A = |A|. (18)

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Complex numbers Matrices

Matrix and linear systems

Consider a known vector b ∈ Rn, a matrix A ∈ Rn×n and unknown x ∈ Rn such that Ax = b. Are there solutions? if A is non-singular, then there exists a unique solution x = A−1b if rank(A) < n there might be no solutions or even infinite of them.

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Complex numbers Matrices

Matrix eigenvalues and eigenvectors

The characteristic polynomial of a square matrix A ∈ Rn×n is the monic1 polynomial pA(λ) |λI − A| = λn + an−1λn−1 + · · · + a1λ + a0, (19) with real coefficients ai. The n roots λi ∈ C of pA(λ) are called eigenvalues of the matrix A. The eigenvalues λi of the matrix A are the only scalars such that rank(λiI − A) = rank(A − λiI) < rank(A). The set of all distinct eigenvalues of A are called the spectrum of A, noted as σ{A}. Each eigenvalue λi of the matrix A is such that Avr

i = vr i λi,

(vl

i)′A = (vl i)′λi

(20) where vr

i ∈ Cn (vl i) is the right(left)-eigenvalue of the matrix A associated to the

eigenvalue λi. Exercise...

1The coefficient of the highest degree is equal to one. 20 / 20