linear algebra. session 2 - math.tamu.eduroquesol/math_304_spring_2018_session_2_print.pdf20 20i 2...
TRANSCRIPT
Matrices. Matrix Algebra
Linear Algebra. Session 2
Dr. Marco A Roque Sol
01 / 23 / 2018
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Applications of systems of linear equationsProblem 2.1.
Find the point of intersection of the lines x − y = −2 and2x + 3y = 6 .SolutionThe intersection point is the solution of the linear system{
x − y = −22x + 3y = 6
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Problem 2.2.
Find the point of intersection of the planes x − y = −2,2x − y − z = 3, and x + y + z = 6SolutionThe intersection point is the solution of the linear system
x − y = −22x − y − z = 3x + y + z = 6
Problem 2.3.
Find a quadratic polynomial p(x) such that p(1) = 4, p(2) = 3,and p(3) = 4
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Solution
Suppose that p(x) = ax2 + bx + c , then
p(1) = a + x + c
p(2) = 4a + 2b + c
p(3) = 9a + 3b + c
The values for a, b, c are given by the solution of the linear systema + b + c = 4
4a + 2b + c = 39a + 3b + c = 4
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Problem 2.4.
Electrical network . Determine the amount of current in eachbranch of the network.
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Solution
To solve this problems, we will use three fundamental Laws comingfrom Physics, namely,
Kirchhofs law 1 ( Charge Conservation ):
At every node the sum of the incoming currents equals the sum ofthe outgoing currents.
Kirchhofs law 2 ( Energy Conservation ):
Around every loop the algebraic sum of all voltages is zero.
Ohm’s Law:
For every resistor the voltage drop E , the current i , and theresistance R satisfy E = iRDr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Thus, applying these three laws to the above circuit we have
Node A: i1 + i2 = i3
Node B: i3 = i1 + i2
Left loop: 10− 10i1 − 40i3 = 0
Right loop: 20− 20i2 − 40i3 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
In this way we have the system:
i3 − i1 − i2 = 0
10− 10i1 − 40i3 = 020− 20i2 − 40i3 = 0
Problem 2.5.
Trafic Flow . Determine the amount of traffic between each ofthe four intersectionsof of the following diagram
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
650 400
610 −→ A x1 −→ B 640 −→
x4 x2
←− 520 D ←− x3 C ←− 600
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Solution
At each intersection, the incoming traffic has to match theoutgoing traffic.
Intersection A : x4 + 610 = x1 + 450
Intersection B : x1 + 400 = x2 + 640
Intersection C : x2 + 600 = x3
Intersection D : x3 = x4 + 520
Which is equivalent to the system:x4 − x1 + 160 = 0x1 − x2 − 240 = 0x2 − x3 + 600 = 0x3 − x4 − 520 = 0
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Matrices
Let us start by solving an m × n system of linear equations
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
where aij are given coefficients, b′ms are given right-hand side, andx ′ns are the unknowns. In this way, we can introduce new arrays ofnumbers to study the linear system
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
In this way, we have the following
Definition
An m× n matrix A , is an array of complex numbers ( m-rows andn-columns ), denoted by
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n
In this context, an element in the i-row and j-column is of thematrix A denoted by aij .
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
An n-dimensional vector v, can be represented as a 1× n matrix(row vector) or as an n × 1matrix (column vector):
v =(x1, x2, x3, · · · , xn
)−→
(x1 x2 x3 · · · xn
)
v =(x1, x2, x3, · · · , xn
)−→
x1x2x3· · ·xn
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
An m × n matrix A = (aij) can be regarded as a column ofn-dimensional row vectors or as a row of m-dimensional columnvectors:
A =
v1
v2
v3
· · ·vm
, vi =(ai1 ai2 ai3 · · · ain
),
A =(
w1 w2 w3 · · · wn
), wj =
a1ja2ja3j· · ·amj
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Associated with any m × n A matrix, we have the following basicmatrices:a) Transpose
Is the ( n ×m ) matrix, denoted by AT , and defined by
AT =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(aTij
)n×m
= (aji )m×n
b) Complex Conjugate
Is the ( m × n ) matrix, denoted by A, and defined by
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
= (aij)m×n = (aij)m×n
c) Adjoint
Is the ( m × n ) matrix, denoted by A∗ = AT
, and defined by
A∗ =
a11 a21 . . . am1
a12 a22 . . . am2...
a1n a2n . . . amn
=(a∗ij)n×m = (aji )m×n
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Basic Matrix Operations
Let A = (aij)m×n and B = (bij)m×n be two matrices, then wedefine
1) A = B ⇐⇒
aij = bij ; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Addtion
A± B = (aij ± bij)m×n
e) Scalar Multiplication
rA = (raij)m×n
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
In particular we have the case of
Vector algebraLet
a =(a1, a2, a3, · · · , an
)and
b =(b1, b2, b3, · · · , bn
)be n-dimensional vectors, and r ∈ R
Vector sum
a + b =(a1 + b1, a2 + b2, a3 + b3, · · · , an + bn
)Scalar multiple
ra =(ra1, ra2, ra3, · · · , ran
)Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Zero vector
r0 =(
0, 0, 0, · · · , 0)
Negative of a vector
−a =(−a1,−a2,−a3, · · · ,−an
)Vector difference
a− b =(a1 − b1, a2 − b2, a3 − b3, · · · , an − bn
)
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Given n-dimensional vectors, {v1, v2, v3, · · · , vk} and scalars{r1, r2, r3, · · · , rk}, the expression
r1v1 + r2v2 + r3v3 + · · ·+ rkvk
is called a linear combination of vectors v1, v2, v3, · · · , vk.
Also, vector addition and scalar multiplication are called linearoperations
Definition. The dot product of n-dimensional vectors
x =(x1, x2, x3, · · · , xn
)and y =
(y1, y2, y3, · · · , yn
)is given by
x · y = x1y1 + x2y2 + · · ·+ xnyn
The dot product is also called the scalar product .
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Review of Matrices
Matrix Multiplication
Let A and B, m × p and p × n matrices respectively
AB = (cij)m×n
where
cij =
p∑k=1
aikbkj
(AB)ij = cij =
. . . . . .. . . . . .ai1 ai2 . . . ain
. . ....
. . . b1j . . .
. . . b2j . . .
. . .... . . .
. . . bnj . . .
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Review of Matrices
That is, matrices are multiplied row by column :
(∗ ∗ ∗ ∗∗ ∗ ∗ ∗
)∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗
=
(∗ ∗ ∗∗ ∗ ∗
)
2× 4 4× 3 2× 3
From another point of view, we have that the matrices A and Bcan be seen as
A =
a11 a12 · · · a1na21 a22 · · · a2n
...
am1 am2 · · · amn
=
v1
v2...
vm
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Review of Matrices
B =
b11 b12 · · · b1pb21 b22 · · · b2p
...bn1 bn2 · · · bnp
=(
w1, w2, . . . , wm
)
⇒
AB =
v1 ·w1 v1 ·w1 · · · v1 ·wp
v2 ·w1 v2 ·w1 · · · v2 ·wp...
vm ·w1 vm ·w1 · · · vm ·wp
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
OBS
In general, when AB is defined, not necessarily BA is also defined,but even in that case, we have in general
AB 6= BA
Example 2.1
Let A and B the matrices defined by
A =
1 −2 10 2 −12 1 1
B =
2 1 −11 −1 02 −1 1
Find A + B, A− B, 3A AB, BA
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Solution
A + B =
3 −1 01 1 −14 0 2
A− B =
−1 −3 2−1 3 −10 2 0
3A =
6 3 −3−3 3 06 −3 3
AB =
1 −2 10 2 −12 1 1
2 1 −11 −1 02 −1 1
=
2 2 00 −1 −17 0 −1
BA =
2 1 −11 −1 02 −1 1
1 −2 10 2 −12 1 1
=
0 −3 01 0 24 −5 4
6= AB
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Example 2.2
Let C and D the matrices defined by
C =
2 11 −12 −1
D =
(1 −2 10 2 −1
)Find CD and DC.Solution
CD =
2 11 −12 −1
(1 −2 10 2 −1
)=
2 −2 11 −4 22 −6 3
DC =
(1 −2 10 2 −1
) 2 11 −12 −1
=
(2 20 −1
)6= CD
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Example 2.3
Using matrix operations rewrite the linear system
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
in terms of matrices.
Solution
Starting with the system
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
a11x1 + a12x2 + . . .+ a1nxn = b1a21x1 + a22x2 + . . .+ a2nxn = b2
......
am1x1 + am2x2 + . . .+ amnxn = bm
and choosing
A =
a11 a12 . . . a1na21 a22 . . . a2n
...am1 am2 . . . amn
X =
x1x2...xm
B =
b1b2...bm
we get
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
AX =
a11x1 + a12x2 + . . .+ a1nxna21x1 + a22x2 + . . .+ a2nxn
...am1x1 + am2x2 + . . .+ amnxn
=
b1b2...bm
= B =⇒ AX = B
Example 2.4
(x1, x2, x3, · · · , xn
)
y1y2y3· · ·yn
=n∑
k=1
xkyk = x · y
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
y1y2y3· · ·yn
( x1, x2, x3, · · · , xn)
=
y1x1 y1x2 · · · y1xny2x1 y2x2 · · · y2xn
...ynx1 ynx2 · · · ynxn
Example 2.5
(1 1 −10 2 1
) 0 3 1 1−2 5 6 01 7 4 1
=
(−3 1 3 0−3 17 16 1
)
2× 3 3× 4 2× 4
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
0 3 1 1−2 5 6 01 7 4 1
( 1 1 −10 2 1
)=
3× 4 2× 4 undefined
Properties of matrix multiplication:
(AB)C = A(BC ) (associative law)
(A + B)C = AB + AC (distributive law 1)
C (A + B) = CA + CB (distributive law 2)
(rA)B = A(rB) = r(AB) (associative law)
Any of the above identities holds provided that matrix sums andproducts are well defined.
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Types of Matrices An m × n matrix A = (aij)m×n is a
1) Zero matrix if aij = 0; i = 1, 2, ...,m, j = 1, 2, ..., n
2) Square Matrix if m = n.
A =
2 −2 11 −4 22 −6 3
; B
(3 75 −4
)
3) Identity matrix (n× n) (In) if aij = δij where δij =
{1 i = j0 i 6= j
A = In =
1 0
1. . .
0 1
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
I1 = (1)
I2 =
(1 00 1
)
I3 =
1 0 00 1 00 0 1
I4 =
1 0 0 00 1 0 00 0 1 00 0 0 1
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
4) Symetric Matrix (n × n) if AT = A or aij = aji ;i = 1, 2, ...,m, j = 1, 2, ..., n
5) Triangular Matrix (n × n)
5a) Upper Triangular Matrix (U) if uij = 0, i > j
U =
a11 · · · · · · a1n
a22. . .
...
0 ann
5b) Lower Triangular Matrix (L) if lij = 0, i < j
L =
a11
a22 0. . .
... · · · · · · ann
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
6) Diagonal Matrix (n × n) (D) if aij = Dij where Dij = diδij
D =
d1 · · · · · ·
...
d2 00 . . .
... · · · · · · dn
Notation
A diagonal matrix D is going to be denoted byD = diag(d1, d2, · · · , dn)
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Diagonal matrices
Theorem
Let A = diag(s1, s2, · · · , sn), B = diag(t1, t2, · · · , tn)then
A + B = diag(s1 + t1, s2 + t2, · · · , sn + tn)
rA = diag(rs1, rs2, · · · , rsn)
AB = diag(s1t1, s2t2, · · · , sntn)
(AB = BA, diagonal matrices always commute)
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Theorem
Let D = diag(d1, d2, · · · , dm) and A be an m × n matrix. Thenthe matrix DA is obtained from A by multiplying the ith row by difor i = 1, 2, ...,m
A =
v1
v2...
vm
⇒ DA =
d1v1
d2v2...
dmvm
Thus, for instance we have
7 0 00 1 00 0 2
a11 a12 a13a21 a22 a23a31 a32 a33
=
7a11 7a12 7a13a21 a22 a23
2a31 2a32 2a33
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Theorem
Let D = diag(d1, d2, · · · , dn) and A be an m × n matrix. Then thematrix AD is obtained from A by multiplying the jth column by djfor j = 1, 2, ..., n
A =(
w1,w2, · · · ,wn
)⇒ DA =
(d1w1, d2w2, · · · , dnwn
)Thus, for instance we have
a11 a12 a13a21 a22 a23a31 a32 a33
7 0 00 1 00 0 2
=
7a11 a12 2a137a21 a22 2a237a31 a32 2a33
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
7) Invertible Matrix (n × n) If A is a square matrix (n × n) andthere exists an n × n matrix B such that
AB = BA = In
The matrix B is denoted by A−1 and is called the Inverse Matrixand A is called invertible or nonsingular matrix. Matrices that donot have an inverse are called singular or noninvertible.OBS
A−1 is the notation for the inverse of A, but keep in mind that
A−1 6= 1
A
A−1A = AA−1 = In
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Example 2.6
A =
(1 10 1
), B =
(1 −10 1
), C =
(−1 00 1
)
AB =
(1 10 1
) (1 −10 1
)=
(1 00 1
)
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
BA =
(1 −10 1
) (1 10 1
)=
(1 00 1
)
C 2 =
(−1 00 1
) (−1 00 1
)=
(1 00 1
)
Thus A−1 = B, B−1 = A, and C−1 = C
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
Inverse matrix
Let Mn(R) denote the set of all n × n matrices with real entries.We can add, subtract, and multiply elements of Mn(R).However, eventhough, we know that the division does not exist ingeneral, for a subset of Mn(R) can be defined as follow. If A andB are n × n matrices and B is an invertible matrix, then
A/B := AB−1
Basic properties of inverse matrices
1) If B = A−1, then A = B−1. In other words, if A is invertible, sois A−1, and A = (A−1)−1 .
Dr. Marco A Roque Sol Linear Algebra. Session 2
Matrices. Matrix Algebra
Matrices, matrix algebra
2) The inverse matrix (if it exists) is unique. Moreover, ifAB = CA = I for some n × n matrices B and C , thenB = C = A−1.
(B = IB = (CA)B = C (AB) = CI = C )
3) If the n × n matrices, A, B, are invertible, so is AB, and(AB)−1 = B−1A−1
4) Similarly (A1A2 · · ·Ak)−1 = A−1k Ak−1 · · ·A−12 A−11
Dr. Marco A Roque Sol Linear Algebra. Session 2