E 600 Chapter 1: Introduction to Matrix Algebra · Chapter 1: Introduction to Matrix Algebra Simona Helmsmueller August 9, 2017. PreviewIntroduction The vector space Mn m Matrices
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Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
E 600
Chapter 1: Introduction to Matrix Algebra
Simona Helmsmueller
August 9, 2017
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Goals of this lecture:
• Be able to apply basic algebraic operations and to name
special matrices
• Be able to apply the GaußJordan algorithm
• Know how to calculate the determinant of a square matrix
and why it is important in the context of linear systems of
equations
• Have a (albeit vague) geometric intuition for the rank
condition
• Know the definition of definiteness of a matrix
• Know the definition of eigenvectors and -values
Following lectures (both in this class and other
courses) will assume these goals have been reached!
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Introduction
x1 + x2 + x3 = 6
x2 − x3 = 0
x1 + x2 + x3 = 3
2x1 = 6− 2x2 − 2x3
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Introduction
x1 + x2 + x3 = 6
x2 − x3 = 0
x1 + x2 + x3 = 3
2x1 = 6− 2x2 − 2x3
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Introduction
x1 + x2 + x3 = 6
x2 − x3 = 0
x1 + x2 + x3 = 3
2x1 = 6− 2x2 − 2x3
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
1. Does a solution exist?
2. Is the solution unique / how many solutions are there?
3. How can we (or our computer) efficiently derive at the (set
of) solutions?
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
An×m =
a1,1 a1,2 · · · a1,ma2,1 a2,2 · · · a2,m
......
. . ....
an,1 an,2 · · · an,m
= (ai ,j)i=1,...,n;j=1,...,m
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Equality of Matrices)
Two matrices are said to be equal if and only if:
(i) they have the same dimension
(ii) their corresponding elements are equal
In symbols, if An×m = (ai ,j)i=1,...,n;j=1,...,m and
Br×s = (bi ,j)i=1...r ,j=1...s , then
A = B ⇔ (n = r, m = s, and, ∀ i = 1, ...,n(= r),
j = 1, ...,m(= s) ai,j = bi,j).
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Addition of Matrices)
For two matrices of identical dimension, we define addition of
matrices in terms of addition of their corresponding elements. In
symbols, if An×m = (ai ,j)i=1,...,n;j=1,...,m and
Bn×m = (bi ,j)i=1,...,n;j=1,...,m, then their sum, denoted by A + B, is
A + B = (ai ,j + bi ,j)i=1,...,n;j=1,...,m.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Multiplication of a Matrix by a Scalar)
Let λ ∈ R. Then to multiply a matrix by this scalar, we multiply
each element of the matrix by this scalar. In symbols, let
An×m = (ai ,j)i=1,...,n;j=1,...,m be any matrix, then:
λA := (λai ,j)i=1,...,n,j=1,...,m
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Vector Space of Matrices)
The set of all n ×m matrices, Mn×m together with the algebraic
operations matrix addition and multiplication with a scalar as
defined above defines a vector space.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Row and Column Vectors) Matrices containing only one row
(n = 1) are called row vectors, and similarly, matrices containing
only one column (m = 1) are called column vectors. In matrix
algebra, if not specified otherwise, by vector we conventionally
mean a column vector.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Square Matrix)
A matrix of size n ×m is said to be a square matrix if and only if
n = m.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Diagonal Matrix)
A square matrix is said to be a diagonal matrix is all of its
off-diagonal elements are zero.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Upper and Lower Triangular Matrix)
An upper triangular matrix is square and has characteristic
elements ai ,j equal to zero whenever i > j and a lower triangular
matrix is square and has characteristic elements ai ,j equal to zero
whenever i < j . Less formally, in the same way that we call a
matrix diagonal if non-diagonal entries are restricted to be null, we
call upper- (lower-) diagonal a matrix whose entries that do not
belong to the part above (below) the diagonal are restricted to be
null.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Symmetric Matrix)
A square matrix is said to be a symmetric matrix if and only if
ai ,j = aj ,i∀1 ≤ i , j ≤ n.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Identity Matrices)
The n × n identity matrix, denoted In, is a diagonal matrix with all
its diagonal elements equal to 1.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Zero Matrices)
The n × n zero matrix, denoted 0n×n, is a square matrix with all
its elements equal to 0.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Transpose of a matrix)
The transpose of a matrix is obtained by reflecting all elements of
a matrix over its main diagonal. In symbols, let
An×m = (ai ,j)i=1,...,n;j=1,...,m be any matrix, then its transpose,
denoted A′m×n or ATm×n, is such that:
A′m×n := (aj ,i )j=1...m,1=1...n
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Product of Matrices)
For two conformable matrices, their product is defined as the
matrix with ijth element equal to the inner product of the ith row
of the first matrix and the jth column of the second matrix. In
symbols, if An×m = (ai ,j)i=1,...,n;j=1,...,m is an n ×m matrix and
Bm×k = (bi ,j)i=1...m,j=1...k is an m × k matrix, the product of
An×m and Bm×k is the n × k matrix Cn×k with characteristic
element:
ci ,j =m∑l=1
ai ,lbl ,j
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Associativity and Distributivity of the Product)
The product for matrices is:
(i) Associative: (AB)C = A(BC)
(ii) Distributive over matrix addition: A(B + C) = AB + AC and
(A + B)C = AC + BC
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Transposition, sum, and product)
(i) If A and B are n ×m matrices, then:
(A + B)′ = A′ + B′
(ii) If A is an n ×m matrix and B is an m × k matrix, then:
(AB)′ = B′A′
(iii) If A is 1× 1 matrix, then A is actually a scalar and A′ = A.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Inverse Matrix)
Let A be an n × n matrix. If there exists a matrix A−1 such that:
A−1A = AA−1 = In
Then A is said to be invertible and A−1 is called the inverse matrix
of A.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
TheoremA square matrix A can have at most one inverse.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
TheoremLet An and Bn be invertible. Then the following holds:
1. (AT )−1
= (A−1)T
2. AB is invertible and (AB)−1 = B−1A−1.
3. for any scalar λ 6= 0, λA is invertible and (λA)−1 =1
λA−1.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
TheoremIf a square matrix An is invertible, then the unique solution to the
system of linear equations Anx = b is x = A−1b.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
The Gauß- Jordan algorithm
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
DefinitionLet
An,n =
a1,1 a1,2 · · · a1,na2,1 a2,2 · · · a2,n
......
. . ....
an,1 an,2 · · · an,n
Then
detA =n∑
j=1
a1,jC1,j
Where C1,j := (−1)1+jA1,j is known as the (1, j)th cofactor of A,
and A1,j, known as the (1, j)th first minor, is the determinant of
the (n− 1)× (n− 1) formed out of A by deleting the first row and
the jth column.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Determinant of the Product)
For any two n× n matrices A and B we have
det(AB) = det(A)det(B).
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
TheoremLet A be a square matrix. Then
A−1 exists ⇔ det(A) 6= 0.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Rank of a Matrix)
Let A be an n ×m matrix. Then, the rank of A, Rank(A), is
defined as the number of linearly independent columns of A. If
n = m and Rank(A)=n, then we say that A has full rank.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Rank condition)
If A has full rank1, then for any b in Rn, the system Ax = b has a
unique solution.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Column Space)
The subset of Rn spanned by the columns of a matrix
A = [a1, ..., an] is called the column space of A:
Col(A) = Span(a1, ..., an).
Theorem
dimCol(A) = rankA
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Column Space)
The subset of Rn spanned by the columns of a matrix
A = [a1, ..., an] is called the column space of A:
Col(A) = Span(a1, ..., an).
Theorem
dimCol(A) = rankA
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
TheoremLet A be n ×m matrix. Then:
1. The system of linear equations represented by Ax = b has a
solution for a particular b ∈ Rn if and only if b ∈ Col(A).
2. The system of linear equations represented by Ax = b has a
solution for every b ∈ Rn if and only if rankA = n.
3. If the system of linear equations represented by Ax = b has a
solution for every b ∈ Rn, then:
n = rankA ≤ number columns of A = m.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Linear Function)
Let X and Y be two vector spaces. A function f : X→ Y is said to
be linear if and only if:
f
(n∑
i=1
λixi
)=
n∑i=1
λi f (xi ) ∀n ∈ N, λi ∈ R, and xi ∈ X .
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Definiteness of a Matrix)
A (symmetric) n × n matrix A is:
(i) positive semidefinite if and only if x ′Ax ≥ 0 for all x ∈ Rn.
(ii) negative semidefinite if and only if x ′Ax ≤ 0 for all x ∈ Rn.
(iii) positive definite if and only if x ′Ax > 0 for all x ∈ Rn.
(iv) negative definite if and only if x ′Ax < 0 for all x ∈ Rn.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition((Leading) principal minors of a matrix)
Let A be an n × n matrix.
(i) Any k × k submatrix of A formed by deleting n − k rows and
the corresponding n − k columns is called a kth order principal
submatrix of A. The determinant of the kth order principal
submatrix of A is called a kth order principal minor of A.
(ii) The k × k submatrix of A formed by deleting the last n − k
rows and columns of A is called the kth order leading principal
submatrix of A, denoted by Ak . Its determinant is called the kth
order leading principal minor of A, denoted |Ak |.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Definiteness of a matrix)
Let A be an n × n symmetric matrix. Then,
(i) A is positive semidefinite if and only if all its principal minors
are non negative.
(ii)A is positive definite if and only if all its n leading principal
minors are strictly positive.
(iii)A is negative semidefinite if and only if, for all k , its kth order
principal minor have the same sign as (−1)k or are 0. (Put in
other words: A is negative semidefinite if and only if every
principal minor of odd order is ≤ 0 and every principal minor of
even order is ≥ 0.)
(iv) A is negative definite if and only if, for all k, its kth order
leading principal minor have the same sign as (−1)k .
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Definition(Eigenvectors and -values)
Let A be a square n× n matrix, then, a vector x ∈ Rn is said to be
an eigenvector of A if and only if there exists a λ in R such that:
Ax = λx
λ is then called an eigenvalue of A.
Preview Introduction The vector space Mn×m Matrices and systems of linear equations Linear functions and matrices
Theorem( Definitness of a Matrix)
Let A be an n × n symmetric matrix. Then:
(i) A is positive definite if and only if all its n eigenvalues are
strictly positive.
(ii) A is negative definite if and only if all its n eigenvalues are
strictly negative.
(iii) A is positive semidefinite if and only if all its n eigenvalues are
non-negative.
(iv) A is negative semidefinite if and only if all its n eigenvalues are
non-positive.
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