Download EGR2013 Tutorial 8 Linear Algebra Outline Powers of a Matrix and

EGR2013 Tutorial 8
Linear Algebra
Outline
z Powers of a Matrix and Matrix Polynomial
z Vector Algebra
z Vector Spaces
Powers of a Matrix and Matrix Polynomial
If A is a square matrix, then we define the nonnegative integer power of A to be
A0 = I
An = AA
A
If A is invertible, then we define the negative integer powers to be
A− n = ( A−1 )n = A−1 A−1
A −1
If A is a square matrix, we have a matrix polynomial in A as
p ( A) = a0 I + a1 A + a2 A2 +
+ an An
Example 1
If A is a n-th order square matrix and Ak = 0 , evaluate ( I − A) −1 .
Solution:
∵ Ak = 0
∴ I − Ak = I
∵ I − Ak = ( I − A)( I + A + A2 +
∴ ( I − A) −1 = I + A + A2 +
1
+ Ak −1 ) = I
+ Ak −1
Definition of Vectors
We often use two kinds of quantities, namely scalars and vectors.
A scalar is a quantity that is determined by its magnitude;
A vector is a quantity that is determined by both its magnitude and its direction.
Equality of Vectors: two vectors a and b are equal, if they have the same
length and the same direction.
Representations: in Cartesian coordinate system, the vector can be described
using real numbers. If the given vector a has an initial point P1 ( x1 , y1 , z1 ) and
a terminal point P2 ( x2 , y2 , z2 ) , then the vector a can be described as
→
a = PP
1 2 = [ x2 − x1 , y2 − y1 , z2 − z1 ]
And the norm a = a12 + a22 + a32 .
Another representation of vectors is a = [a1
a3 ] = a1i + a2 j + a3k ,
a2
where i, j, k are the standard unit vectors in the positive directions of the axes
in a Cartesian coordinate system.
Basic Properties of Vectors:
¾ Zero vector 0 has length 0 and no direction.
¾ Negative vector –a has the length a and the direction is opposite to that
of a.
¾ Unit vector is a vector of norm 1
Vector Addition and Scalar Multiplication
If a = [a1
a2
a3 ] and b = [b1 b2
b3 ] are two vectors, then
a ± b = [a1 ± b1 a2 ± b2
ka = [ka1 ka2
Basic Properties:
¾ a+b=b+a
¾ a+0=0+a=a
¾ k(a+b)=ka+kb
¾ k(la)=(kl)a
¾ 1a=a
a3 ± b3 ]
ka3 ]
(a+b)+c=a+(b+c)
a+(-a)=0
(k+l)a=ka+la
0a=0
(-1)a=-a
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Inner Product or Dot Product
Definition: if a and b are two vectors and θ is the angle between a and b,
then the dot product or Euclidean inner product a ib is defined by
a ib = a b cosθ = a1b1 + a2b2 + a3b3
Angle between two vectors:
The angle θ between the vectors is cosθ =
a ib
.
a b
Orthogonality:
The inner product of two vectors is zero if and only if these vectors are
perpendicular.
Properties of the Dot Product:
¾
Schwarz inequality:
¾ Triangle inequality:
a ib ≤ a b
a ib ≤ a + b
Orthogonal Projection:
Orhtogonal projection of a vector a in the direction of a vector b is defined by
p = projb a =
a ib
b
2
b
Example 2
If vector a is perpendicular to any vector, show a is a zero vector.
Proof:
∵ a is perpendicular to any vector
∴for any vector β , a i β = 0
especially, if β = a , thus a ia = 0
∴a =0
therefore, a is a zero vector.
Zero vector is perpendicular to any vector.
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Example 3
Show vector c and vector (a ic)b-(bic)a are orthogonal vectors.
Proof:
[(a ic)b-(bic)a]ic
=[(a ic)b]ic-[(bic)a]ic
=(a ic)(bic)-(bic)(a ic)
=0
Thus c and (a ic)b-(bic)a are orthogonal vectors.
Vector Product or Cross Product
Definitions: if a and b are two vectors, then the cross product or vector product
is a vector v = a × b = [v1
v2
v3 ]
The vector v can be obtained from the expansion by the first row of the
symbolical third-order determinant
i
v= a1
b1
j
a2
b2
k
a
a3 = 2
b2
b3
a 3 a1
ib3 b1
a3
a
j+ 1
b1
b3
a2
k
b2
The length of the vector v is given as a × b = a b sinθ
The direction of the vector v is perpendicular to both a and b.
Example 4
If two vectors a and b, a = 10 , b = 2 and a ib = 12 ,
Solution:
∵ a ib= a b cos θ
∴ cos θ =
∴ sin θ =
aib 3
=
a b 5
4
5
∴ a × b = a b sin θ = 16
4
Evaluate a × b .
Example 5
Given a × b=c × d,a × c=b × d , show a-d is parallel to b-c .
Proof:
If a-d is parallel to b-c , then (a-d) × (b-c) should be 0.
∴ (a-d) × (b-c)=a × b+d × c-d × b-a × c
=a × b-c × d + b × d-a × c=0
therefore, vector a-d is parallel to vector b-c .
Scalar Triple Product
If a, b and c are three vectors, then the scalar triple product is defined by
a1
a i(b × c)= b1
c1
a2
b2
c2
a3
b3
c3
The absolute of the scalar triple product is the volume of the parallelepiped
with a, b and c as edge vectors.
Example 6
Given three vectors a=i+4j-4k, b=-5i+5j+k and c=-6i+j+5k, show these three
vector are in the plane.
Proof:
Because the absolute of the scalar triple product is the volume of the
parallelepiped, if the volume is zero, we can say the edge vectors are in the
same plane.
a1
Thus, a i(b × c)= b1
c1
a2
b2
c2
a 3 1 4 -4 1 4 -4
b3 = -5 5 1 = 0 25 -19 = 0
c3 -6 1 5 0 25 -19
Therefore, vectors a, b and c are in the same plane.
™ If a ib = 0 , then vector a is perpendicular to vector b;
™ If a × b = 0 , then vector a is parallel to vector b;
™ If a i(b × c)=0 , then vectors a, b and c are in the same plane.
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Vector Spaces
In a nonempty set V, we define two algebraic operations:
(1) vector addition: ∀α , β ∈ V ⇒ α + β ∈ V
(2) scalar multiplication: ∀α ∈ V , ∀k ∈ R ⇒ kα ∈ V
For vector addition:
¾ a+b=b+a
¾ (a+b)+c=a+(b+c)
¾ there is an object 0 in V, such that 0+a=a for all a in V
¾ for each a in V, there is an object –a in V, such that a+(-a)=0
For scalar multiplication:
¾ k(a+b)=ka+kb
¾ (k+l)a=ka+la
¾ k(la)=(kl)a
¾ for every a in V, 1a=a
If V follows all the axioms above, then V is called a vector space.
Example 6
Let V = R
2
and we define the addition and scalar multiplication in V for two
vectors a = [ a1
a2 ] and b = [b1 b2 ] and a scalar k as follows:
a + b = [a1
a2 ] + [b1 b2 ] = [a1 + b1 0]
ka = k[a1
First,
∀α , β ∈ V ⇒ α + β ∈ V
a2 ] = [ka1 0]
and
∀α ∈ V , ∀k ∈ R ⇒ kα ∈ V
satisfied.
But there is no zero vector in V, such that [ a1
a2 ] + [? ?] = [a1
So set V is not a vector space with stated operations.
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a2 ]
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