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Chapter 1 Linear Algebra

Page history last edited by Louis 13 years, 5 months ago

Linear Algebra


1 Basic concepts of matrices


1.1 Definitions of Matrices


A matrix is a rectangular array of numbers enclosed in brackets.

The numbers in the array are called the entries or elements of the matrix.
The size of a matrix is described in terms of the number of rows (horizontal lines) and columns (vertical lines) it contains.
A general Formula matrix has a from


                                                            Formula


Formula: the number of rows; Formula: the number of columns.
Note. If Formula, we call Formula an Formula square matrix. Its diagonal containing the entries Formula is called the main diagonal of Formula.
A matrix that is not square is called a rectangular matrix.


Example 1.1.

 

Some examples of matrices are

 

                                                            Formula


The size of the above matrices are Formula (read 3 by 2), Formula and Formula.
A matrix with only one column is called a column matrix (or a column vector), for example, the Formula matrix

 

                                                            Formula

 

A matrix with only one row is called a row matrix (or a row vector), e.g. the Formula matrix


                                                            Formula

 

1.2 Equality of matrices


Two matrices Formula and Formula are equal, if and only if they have the same size and the corresponding entries are equal, i.e. Formula and so on.


Example 1.2.

 

Consider the matrices


                                                            Formula


If Formula, then Formula. But for all other values of Formula, the matrices Formula and Formula are not equal since not all of their corresponding entries are equal. There is no value of Formula for which Formula since Formula and Formula have different sizes.

 

Exercise 1.2.

 

Consider


                                                            Formula


and Formula. Find Formula.

 

2 Basic matrix operations


2.1 Matrix addition and subtraction


If Formula and Formula are matrices of the same size, then the sum Formula is the matrix obtained by adding the entries of Formula to the corresponding entries of Formula. Similarly, the difference Formula is the matrix obtained by subtracting the entries of Formula from the corresponding entries of Formula.
Note. Matrices of different sizes cannot be added or subtracted.


Example 2.1.

 

Consider the matrices


                                                            Formula

 

Then


                                                            Formula


The expressions Formula, Formula, Formula and Formula are undefined because the matrices are of different sizes.
The matrix addition or subtraction has the following properties

 

  • Formula
  • Formula
  • Formula (commutative law)
  • Formula (associated law)


Here Formula is zero matrix of size Formula, that is, a matrix with all entries zero.

 

Exercise 2.1.


                                                            Formula


Find Formula and Formula.

 


2.2 Scalar Multiplication


If Formula is any matrix and Formula is any scalar, then the product Formula is the matrix obtained by multiplying each entry of the matrix Formula by Formula. The matrix Formula is said to be a scalar multiple of Formula.


Example 2.2.

 

Let


                                                            Formula


Then


                                                            Formula


The scalar multiplication has the following properties

 

  • Formula
  • Formula (associated law)
  • Formula (distributive law)
  • Formula

 


2.3 Matrix multiplication


If Formula is an Formula matrix and Formula is an Formula matrix, then the product Formula exists if and only if Formula. Formula is an Formula matrix with entries

 

                                                            Formula

 

That is, to get Formula, multiply each entry in the Formulath row of Formula by the corresponding entry in the Formulath column of Formula and then add these Formula products.


                                                            Formula

 

Note.

  • The number of columns of Formula must be the same as the number of rows of Formula in order to form the product Formula.
  • If this condition is not satisfied, the product is undefined.
  • A convenient way to determine whether a product of two matrices is defined is to write down the size of Formula and, to the right of it, write down the size of Formula.
  • If, as shown in the figure below, the inside numbers are the same, then the product is defined.
  • The outside numbers then give the size of the product.

 

 

Example 2.3.

 

Matrix multiplication

(a) Products of two matrices


                                                            Formula

 

Here Formula is Formula and Formula is Formula, so that Formula comes out Formula, whereas Formula is undefined.
(b) Products of a matrix and a row matrix


                                                            Formula

 

whereas

 

                                                            Formula

 

is undefined.

(c) Products of a row and a column matrix

 

                                                            Formula

 

Note. The matrix multiplication has the following properties

  • In general matrix multiplication is not commutative, i.e. Formula.
  • Formula does not necessarily imply Formula or Formula or Formula.
  • Formula does not necessarily imply Formula (even when Formula).
  • Formula, Formula is any scalar
  • Formula= Formula
  • Formula = Formula
  • Formula = Formula

 


Exercise 2.3.

 

Consider


                                                            Formula


Observe (a), (b), (c) and (e).

 

2.4 Matrix transposition


The transpose Formula of an Formula matrix Formula is the Formula matrix that results from interchanging the rows and columns of Formula. That is, the first column of Formula is the first row of Formula, the second column of Formula is the second row of Formula, and so forth.

 

                                                            Formula


Example 2.4.

 

The following are some examples of matrices and their transposes.


                                                            Formula


The transposition has the following properties

  • Formula = Formula
  • Formula = Formula
  • Formula = Formula
  • Formula = Formula

 


Exercise 2.4.

 

Consider


                                                            Formula


Observe (d).

 

2.5 Matrix trace


If Formula is a square matrix, then the trace of Formula, denoted by Formula , is defined to be the sum of the entries on the main diagonal of Formula. The trace of Formula is undefined if Formula is not a square matrix.


Example 2.5.


                                                            Formula


Exercise 2.5.


                                                            Formula


Since Formula, find Formula.

 

3 Special Matrices


The following are certain kind of matrices that are frequently used.


3.1 Symmetric matrices


Symmetric matrices are square matrices whose transpose equals the matrix, i.e.


                                                            Formula


By inspection, the mirror images of the entries of symmetric matrices across the main diagonal must be equal. The product of any matrix and its transpose, i.e. Formula or Formula will always give symmetric matrices.


Example 3.1.

 

Symmetric matrices
Let Formula be the Formula matrix


                                                            Formula


Observe that Formula and Formula are symmetric matrices.


3.2 Skew-symmetric matrices


Skew-symmetric matrices are square matrices whose transpose equals the negative of the matrix, i.e.


                                                            Formula


The subtraction of a square matrix and its transpose, i.e. Formula will always give skew-symmetric matrices.


Example 3.2.

 

Skew-symmetric matrices
Let Formula be the Formula matrix


                                                            Formula


Observe that Formula is a skew symmetric matrix.

 

3.3 Upper triangular matrices


Upper triangular matrices are square matrices that can have nonzero entries only on and above the main diagonal, whereas any entry below the diagonal must be zero.


Example 3.3.

 

Upper triangular matrices


                                                            Formula

 

3.4 Lower triangular matrices


Lower triangular matrices are square matrices that can have nonzero entries only on and below the main diagonal, whereas any entry above the diagonal must be zero.


Example 3.4.

 

Lower triangular matrices


                                                            Formula


3.5 Diagonal matrices


Diagonal matrices are square matrices that can have nonzero entries only on the main diagonal . Any entry above or below the main diagonal must be zero.
Special kind of diagonal matrices:

 

  • Scalar matrix S is a diagonal matrix with all the diagonal entries equal to Formula. Multiplication of any square matrix Formula of the same size by S has the same effect as the multiplication by a scalar.

 

                                                            Formula

 

  • Identity or unit matrix I is a diagonal matrix with all the diagonal entries equal to 1.

 

                                                            Formula


Example 3.5.

 

Diagonal matrix Formula, scalar matrix Formula, unit matrix Formula

 

                                                            Formula

 

4 Basic concepts of vectors

 

4.1 Definitions of vectors


A vector is a matrix that has only one row or one column. If the matrix has one row then we call the matrix a row vector. It is of the form

 

                                                            Formula


If the matrix has one column then we call the matrix a column vector. It is of the form


                                                            Formula

 

The matrix transposition converts a row vector to a column vector and vice versa.


Example 4.1.


                                                            Formula


Formula and Formula are row vectors. Formula and Formula are column vectors.


4.2 Inner product of vectors


If Formula is a row vector and Formula a column vector, both with Formula components, then matrix multiplication (row times column) gives a Formula matrix or a scalar. This product is called the inner product or dot product of Formula and Formula and is denoted by Formula. Thus,


                                                            Formula


Example 4.2.

 

Let Formula and Formula then


                                                            Formula

 

Exercise 4.2.

 

Let Formula and Formula then find Formula.


4.3 Matrix multiplication in terms of row and column vectors


Matrix multiplication Formula is a multiplication of rows of Formula into columns of Formula. Thus, we can write the product in terms of inner products of row and column vectors. This can be explained as follows.
An Formula matrix Formula can be represented by a column matrix which elements are row vectors.


                                                            Formula


An Formula matrix Formula can be represented by a row matrix which elements are column vectors.

 

                                                            Formula

 

Suppose that Formula. Then we have


                                                            Formula

 

and


                                                            Formula


Example 4.3.

 

The row vectors Formula, Formula, Formula of the matrix


                                                            Formula


The column vector Formula, Formula of the matrix


                                                            Formula


Hence from (2) we obtain the product

 

                                                            Formula

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