Chapter 3
Matrix and Matrix Algebra
3.1 Matrix operation
3.2 Inverse matrix
3.3 Elementary matrix
3.4 Subsapce and linear independence
3.5 Solution set of a linear system and matrix
3.6 Special matrices
*3.7 LU-decomposition
Matrix is widely used as a tool to transmit digital sounds and images through internet as well as solving linear systems. We define the addition and product of two matrices.
These operations are tools to solve various linear systems. Matrix product also becomes an excellent tool in dealing with function composition.
In the previous chapter, we have found the solution set using the Gauss elimination method.
In this chapter, we define the addition and scalar multiplication of matrices and introduce algebraic properties of matrix operations.
It will be used to describe the relation between solution set and matrix, Then using the Gauss elimination, we show how to find the inverse matrix.
Furthermore, we investigate the concepts such as linearly independence and subspace which are necessary in understanding the structure of a linear system.
Finally we introduce some interesting special matrices.
3.1 Matrix operation
Reference video: https://youtu.be/C56kVi-AZW8 (http://youtu.be/DmtMvQR7cwA)
Practice site: http://matrix.skku.ac.kr/knou-knowls/CLA-Week-3-Sec-3-1.html
|
|
|
|
This chapter introduces the definition of the addition and scalar multiplication of matrices and the algebraic properties of the matrix operations. Although many of the properties are identical to those of the operations on real numbers, some properties are different. Matrix operation is a generalization of the operation on real numbers. |
|
Definition |
[Equality of Matrices] |
||
|
|
|
|
|
|
|
Two matrices |
|
|
|
|
|
||
|
|
|||
and 
of same size are 
for all 
, and denote it by 
.
To define equal matrices, the size of two matrices should be the same.
For what values of 
the two matrices
, 
are equal?
For 
, each entry should be equal. Thus (that is, 
) 
, 
, 
, 
. ■
|
Definition |
[Addition and scalar multiplication of matrix] |
||
|
|
|
|
|
|
|
Given two matrices
|
|
|
|
|
|
||
|
|
|||
and 
and a real number 
, the 
of 
and 
, and 
of 
by 
are defined by
, 
.
To define addition, the size of two matrices should be the same.
For 
, what is 
, 
, 
?
,
. □
● http://matrix.skku.ac.kr/RPG_English/3-MA-operation.html
● http://matrix.skku.ac.kr/RPG_English/3-MA-operation-1.html
http://sage.skku.edu or http://mathlab.knou.ac.kr:8080
and 
, the 
of 
and 
is defined below.
,
.
and 
to be compatible for multiplication, we require the number of columns of 
to be equal to the number of rows of 
.
is of size number of rows of 
by the number of columns of 
.
, and denote the 
th row of 
by 
and the 
th column of 
by 
. Then
th row vector of 
and the 
th column vector of 
is the 
entry of 
.
, 
. Then 
□
, 
, 
be matrices of proper sizes (oeprations are well defined) and let 
be scalars. Then the following hold.
Since 
, we have
, we have
. 
.
The properties of operations on matrices are similar to those of operations on real numbers which are well known,
, we do not have 
in general.
.
, 
.
is defined but 
is not defined. Similarly 
is a 
matrix but 
is a 
matrix, and hence 
. Also although 
and 
are 
matrices, as we can see below, we have 
.
(or 
).
and a zero matrix 
of a proper size, the following hold.
, it is possible to have 
, 
. Similarly,
, 
, it is possible to have 
.
.
. But 
and 
.
but 
. 
with diagonal entries all 1's is called an 
and is denoted by 
. That is,
Let 
be an 
matrix and the identity matrix 
. It is easy to see that
.
. Then 
,
.
be a square matrix of order 
. The 
th 
is defined by
(
times)
is a square matrix and 
are non negative integers, then
.
In the set of real numbers, we have 
.
.
, we have 
.
, the 
is denoted by 
and defined by
, 
.
The transpose 
of 
is obtained by interchanging the rows and columns of 
.
, 
,
. □
be matrices of appropriate sizes and 
a scalar. The following hold.
.
. Show that (3) of Theorem 3.1.4 is true.
Since 
, 
.
. Thus 
.
is defined by 
.
, 
are square matrices of the same size and 
, then
, 
. 
Reference video: 
Practice site: 
of order 
is called 
such that
.
if exists is called the 
. If such a matrix 
does not exist, 
is called 
. Note that the third row of 
has all zeroes. Thus for any matrix
the third row of 
is 
. Therefore there does not exist 
such that 
, that is, 
is singular.
is an invertible square matrix of order 
, then an inverse of 
is unique.
are inverses of 
. Then as
, 
is unique. 
to be invertible is that 
. Hence one has
are invertible square matrices of order 
and 
is a nonzero scalar, then the following hold.
is invertible and 
.
is invertible and 
.
is invertible and 
.
is invertible and 
.
(2) 
is an invertible matrix, then so is 
and the following holds.
.
. 
.
Since 
,
, we have
. Also since
we have
. 
Reference video: 
Practice site: 
by 
matrix is called an 
by performing a single 
.
: Interchange the 2nd and the 3
: Multiply the 2nd row by 3. 
: 
on the left and any matrix 
is the matrix that results when the corresponding same row operation is performed on 
.
, 
, 
, 
matrix 
, the followings are equivalent.
is invertible.
is row equivalent to 
. (i.e. RREF
)
can be expressed as a product of elementary matrices.
has only the trivial solution 
.
, augment 
on the right side so that we make
matrix 
.
.
be the RREF of 
in the step 2. Then,
, then 
.
, then 
is not invertible so that 
does not exist.
Consider 
. Then
, 
.
, 
does not exist. 
. 
using slicing of the above matrix.
Reference video: 
Practice site: 
.
.
with standard addition and scalar multiplication is also called a vector space over 
and its elements are called vectors.
be a nonempty subset of 
. Then 
is called a 
if 
satisfies the following two conditions.
(
All subspaces of 
contain zero vector.
are subspaces of 
where 
is denoted by the origin. They are called the trivial subspaces.
of 
satisfies two conditions for subspace. Hence, 
is a subspace of 
. On the other hand, a subset 
of 
does not satisfy conditions for subspace so that 
is not a subspace of 
.
but 
are one of the followings.
are one of the following.
is a subspace of
.
, 
, 
is a subspace of 
. 
denote the set of all 
matrices over 
.
, show that
. (This 
is called a 
)
so that 
. Since for 
, 
, 
,
and
.
.
is a subspace of 
. 
can be expressed in the form
, then 
is called a 
.
be vectors of 
. Can 
be a linear combination of 
and 
?
in 
such that
exist. Consequently, 
is not a linear combination of 
. 
.
, 
. Then there exist 
such that
.
,
.
.
is a subspace of 
. 
, we saw that for a subset 
,
of 
.
is a
spanned by 
. We denote it
or 
.
can be expressed a linear combination of 
, then 
spans 
. That is,
is a spanning set of 
.
is a spanning set of 
.
. Then, 
columns 
of 
span a subspace of 
. This subspace is called a column space of 
, denote by
or Col(
).
is defined by a subspace of 
spanned by 
rows 
of 
, denoted by
or Row(
).
spans 
or not.
, 
, 
such that a given vector 
is written as
. 
cannot be determined. Therefore this linear system has a case that the system cannot determine a unique solution. 
satisfies
(or subset 
) are called 
(or subset 
) are not linearly independent, then it is called
If 
is linearly dependent, there exist at least one non-zero scalar
such that
.
The 
.
, 
is linearly independent.
,
, and 
is linearly independent. 
in 
are linearly independent, then
,
.
are linearly independent,
are linearly independent. 
, Show that 
is linearly dependent.
, if 
, then
, 
, 
. Therefore there exist non zero scalars 
, 
is linearly dependent. 
, the followings hold.
is linearly dependent 
can be expressed as a linear combination of the other elements in 
.
contains the zero vector, then 
is a linearly dependent.
is 
is also linearly dependent.
is 
is also linearly independent.
(1) (
) If 
is linearly dependent, then there exist 
such that
is a nonzero.
then,
can be expressed as a linear combination of the other vectors in 
) Without loss of generality, we can write
is linearly dependent since 
.
In other words, that set 
is linearly independent means that any vector in 
cannot be written as a linear combination of the other vectors in 
.
In 
, there are at most 
vectors in a linearly independent set.
, 
vectors are always linearly dependent.
in 
, we can easily check that 
is linearly dependent from Theorem 3.4.2. 
in 
. 
is a subspace containing the zero vector. Also 
forms a 
and parallel to 
. In other words, 
is translate of 
by 
.
are vectors in 
, then the set of vectors 
(
) is a subset of 
which is the translation of a subspace
that pass through the origin, by 
.
Reference video: 
Practice site: 
matrix 
is invertible and 
is a vector in 
, the system
.
.
. It is easy to show that 
is invertible,
.
. 
, where
is called a 
is called a 
equations and 
variables such that 
(
: a real number). Then the solution to (2) is
, and nontrivial if 
. 
, 
is called the 
.
Since the matrix size is greater than 2, let us use Sage.
, 
, 
, 
.
and 
are free variables.
, 
. Then we have
below.
a 
.
and 
, then
.
has solutions. Let 
be a solution space to 
. If 
is a solution to 
, then
.
and 
which is a solution set of 
is a set of 
is added to a solution set 
of 
.
.
matrix 
, the following are equivalent.
is a product of elementary matrices.
is invertible.
is the unique solution to 
.
has a unique solution for any 
.
are linearly independent.
are linearly independent.
are orthogonal to the rows of 
.
with 
variables. If the system has 
linear equations, then the size of matrix 
is 
. It can be rewritten using inner product. Let 
, 
, 
, 
indicate rows of a matrix 
.
(
) if 
is a solution to 
. That is, the vectors in this solution space of 
are all 
.
is orthogonal to row vectors of the coefficient matrix 
.
-plane:
where 
-space:
where 
in 
where 
forms a plane. If 
, then it is a 
for a nonzero vector 
. This solution set 
is called an 
is called a
.
: a normal vector of the hyperplane 
, which is 
= the orthogonal complement of 
.
Reference video: 
Practice site: 
with its main diagonal entries 
can be written as 
, 
and 
are scalar matrices.
and 
and
.
and 
,
.
, 
is obtained by multiplying each row of 
by the corresponding entry of 
,
is obtained by multiplying each column of 
by the corresponding entry of 
,
, 
, 
satisfies 
, 
is called a 
, then 
is called a 
and 
are symmetric matrices and 
is a skew-symmetric matrix.
is a square matrix, prove the following.
is a symmetric matrix.
is a skew-symmetric matrix.
, 
is
, 
is
For any 
and 
is a symmetric matrix and
is a skew-symmetric matrix. 
triangular matrices are as follows.
and 
be a lower triangular matrix.
is a lower triangular matrix.
is an invertible matrix, then 
is a lower triangular matrix.
for all 
, then the main diagonal entries of 
is all 1’s.
be a square matrix. If there exists an positive integer 
such that
(
is called nilpotent), 
is invertible and 
. This is because
Indicate whether the statement is true (T) or false (F). Justify your answer.
and 
in 
is a plane.
and 
, then the three vectors are linearly independent.
that contains is linearly dependent.
, 
, 
} is a linearly independent set, then so is the set {
, 
, 
} for every nonzero scalar 
.
and 
are linear dependent, linear combinations of two vectors are not plane.
= (1, 0), 
= (0, 1), 
= (0, 2), we cannot make 
by linear combination 
, 
. However, 
and 
are linearly dependent.
is only 
, a solution of 
is 
. So {
} are linearly independent.
When 
, confirm the following. 
.
i) 
■
■
When 
, confirm that 
and that 
.
, 
but 
When 
, compute the following.
■
■
Show that 
is the inverse of 
. And confirm that 
.
=
=
=
=
=
=
= 
=
=
=
■
If 
, show that 
.
, then 
. ■
Find a 
elementary matrix corresponding to each elementary operation.
(Elementary matrix) 
: Interchange the 2nd and the 3rd rows 
: Multiply the 2nd row by 2.
: Add –2 times the 1st row to the 3rd row. ■
Using elementary operations, find the inverse of the following matrix.
(2) 
= 
→ 
→ 
=
. 
=
= 
→ 
. 
.
Let 
and 
be any 
matrix.
and confirm how 
affects on 
.
and confirm how 
affects on 
.
= 
, 
= 
= 
affects on the 3rd row of 
.
= 
= 
affects on the 1st column of 
. 
is a subspace of 
.
is closed under the addition.
is closed under the scalar multiplication.
is not a subspace of 
. ■
Determine if 
is a subspace of 
.
is closed under the addition.
is closed under the scalar multiplication.
is a subspace of 
. ■
Find a vector equation and a parameterized equation of the subspace spanned by the following vectors.
, 
, 
(a) 
, 
, 
where 
, 
in ℝ.
, 
, 
, 
, 
. ■
Give a solution by finding the inverse of the coefficient matrix of the system.
Set the coefficient matrix 
.
:
Ans) 
=
■
Determine if the homogeneous system has a nontrivial solutoin.
=
: Augmented matrix
=
: RREF(
)
Check if the following matrix is invertible. If so,
is a diagonal matrix.
.
, 
, 
.
, 
, 
.
is
■
Find the product by using a property of special matrices.
, 
: diagonal matrices
= 
: 
, 
, 
was multiplied on the left.
= 
: 
, 
was multiplied on the right.
. ■
Determine 
so that 
is skew-symmetric matrix.
The matrix 
is a skew-symmetric matrix then 
and 
.
. 
If 
satisfies 
and 
,
can be expressed as follows.
?
=> 
■
Let 
be a square matrix. Explain why the following hold.
contains a row or a column consisting of 0's, 
is not invertible.
contains the same rows or columns, 
is not invertible.
contains a row or column which is a scalar multiple of another row or column of 
.
is not invertible. 
det
=0
contain a row or a column of all zeros)
is not invertible.
has 
, 
which are 
, we can make a new matrix 
on 
. Because the matrix 
has a row or a column
=det
, det
=det
=0. So, 
is not invertible.
is not invertible.
has 
, 
which are 
(
is constant), we can make
which take 
on 
. Because the matrix 
has a
=det
, det
=det
=0. So, 
is not
is not invertible. 
Let 
be an 
square matrix. Discuss what condition is need to have 
.
,
. ■
가 invertible이 아니면 성립하지 않을 수 있다.
Find 
matrices 
, 
and explain the relation with ERO.
■
Decide if the following 4 vectors are linearly independent.
, 
, 
, 
are linearly dependent. ■
If 
and 
have a solution, prove that 
has a solution.
and 
have a solution, prove that 
has a solution.
and 
be solutions of 
and 
respectively.
and 
is a solution of 
and 
have a solution, then 
has a solution. 
Suppose 
is an invertible matrix of order 
.
in 
is orthogonal to every row of 
, what is 
?
For 
in 
is orthogonal to every row of 
,
=0,
=0
=0 
=0
Null(
)
is a solution of 
.
Prove that a necessary and sufficient condition for a diagonal matrix to be invertible is
= 
0 
for all 
(
). ■
If 
is invertible and symmetric, so is 
.
, 
and 
.
=> 
=> 
is symmetric. 
