# Series:

The sum of an indicated number of terms in a sequence is called a series. Series is widely used in calculus and mathematical analysis. Series are also used in many fields like physics statistics finance etc.

## History of series:

• Archimedes  a Greek mathematician produce the first  known summation of an infinite Series with a method that is used in the field of calculus today.He used the method of exhaustion to find the area under the arc of a parabola with the summation of infinite series.
• In 17th century the Mathematician James Gregory work in new decimal system om infinite series and published many Maclaurine series.
• In 1715 the mathematician Brook Taylor gave a general method for constructing the Taylor series for all function for which they exist.

#### Example:

The sum of the first seven terms of the sequence {n²} is the series

1+4+9+16+25+36+49.

The above series is also named as the 7th partial sum of the sequence {n²}.

If the number of terms in a series is finite then the series is called a finite series while a series consisting of an unlimited number of terms is an infinite series.

#### Notation:

The Greek letter E is used to represent the sum of series. This notation is called summation or sigma. For example, the series 3+6+9+12+15 can be represented as 3\overset\infty{\underset{n=1}{\sum a_n}} where the n is called the index of summation.

### Sum of first n terms of an Arithmetic series:

for any sequence {a_n} we have

S_n = a_1 + a_2 +a_3 +….+ a_n

if {an} is an A.P then S_n can be written with usual notation as

S_n = a_1 +( a_1 +d) + ( a_1 + 2d)+ …. +( a_n -2d) +( a_n -d) + a_n (1)

if we write the terms of series in the reverse order the sum of nterms remain the same that is

S_n = a_n +( a_n -d) = ( a_n -2d)+…….( a_1 +2d) = (a_1 +d) + a_1 (2)

adding (1) and (2) we get

2 S_n = ( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n ) =….=( a_1 + a_n ) +( a_1 + a_n ) +( a_1 + a_n )

S_n = ( a_1 + a_n ) +( a_1 + a_n )….to n terms

n( a_1 + a_n )

thus S_n = \frac n2 ( a_1 + a_n )

S_n =\frac n2[ a_1 + a_1 +(n-1)d]

S_n = \frac n2 [2 a_1 +(n-1)d]

## Infinite Series:

Let {an} be a sequence .an expression of the form

a_1 + a_2 +a_3 +….+ a_n +….

containing infinitely many terms of the sequence of {an} is called an infinite series.

this series is symbolically written as.

\overset\infty{\underset{n=1}{\sum a_n}} or \overset\infty{\underset1{\sum a_n}}

here the a_n is called the nth term of the series.

Let S_n denote the sum of the first n terms of this series then we write

S_1 = a_1

S_2 = a_1 + a_2

S_3 = a_1 + a_2 + a_3

.

.

.

S_n = a_1 + a_2 + a_3 +……+ a_n

the sequence { S_n } is called the sequence of partial sums of the upper series and the number S_n is called the nth partial sum of the series \overset\infty{\underset1{\sum a_n}}

### Convergeand Diverge:

Let { S_n } be the sequence of partial sum of the series \overset\infty{\underset1{\sum a_n}} . if the sequence { S_n } converge to the limit s, then \overset\infty{\underset1{\sum a_n}} is said to be converge and S is called the sum of series .

if the sequence diverge then the series is said to be divergent.

To check the convergence of series we may have use some tests. These test are tell us about the convergence or divergence of series. These tests are following.

• The Basic Comparision Test
• The Limit Comparision Test
• The Integral Test
• The Ratio Test
• The Cauchy Test

# Triangular Matrix:

A square matrix in which all the entries upper or below the maim diagonals elements are zero is said to be a triangular matrix. The triangular matrix must be a square matrix that means the triangular matrix has the same number of rows and columns.

There are two types of triangular matrices. the upper triangular matrix and lower triangular matrix

## Types of a Triangular Matrix:

### Upper Triangular Matrix:

A square matrix in which entries below the main diagonals are zero is called an upper triangular matrix.

#### For Example:

\begin{bmatrix}5&3&2&5\\0&7&7&9\\0&0&4&1\\0&0&0&8\end{bmatrix}

### Lower Triangular Matrix:

A square matrix in which the entries above the main diagonals are zero is called a lower triangular matrix.

#### For Examples:

\begin{bmatrix}9&0&0&0\\6&2&0&0\\7&3&8&0\\3&5&9&6\end{bmatrix} \begin{bmatrix}6&0&0\\9&1&0\\7&3&7\end{bmatrix}

Notation

The capital word U is the notation of the upper triangular matrix it is also called the right triangular matrix. And the capital word L is used for the lower triangular matrix it is also called the left triangular matrix.

### Properties of an Upper Triangular Matrix:

• If we added or subtract two upper triangular matrices then the matrix that we obtain is also a upper triangular matrix
• If we multiply two upper triangular matrix the resultant matrix also upper triangular matrix
• If we take inverse of upper triangular matrix then the resultant matrix is upper triangular matrix
• Transpose  of upper triangular matrix is lower triangular matrix such that U^t = L.
• The upper triangular remain unchanged when we multiply it by any scalar quantity .

### Properties of a Lower Triangular Matrix:

• If we added or subtract two lower triangular matrices then the resultant matrix is a lower triangular matrix.
• If we multiply two lower triangular matrix the resultant matrix also lower triangular matrix
• If we take inverse of lower triangular matrix then the resultant matrix is lower triangular matrix
• Transpose  of lower triangular matrix is lower triangular matrix such that L^t</span> = U .
• The lower triangular remain unchanged when we multiply it by any scalar quantity .

Some other types of a triangular matrix

### Unit Triangular Matrix:

In any upper or lower triangular matrix if the main diagonals entries are 1 then it is called unit (upper or lower) triangular matrix. Mostly it is called normed triangular matrix

\begin{bmatrix}1&0&0&0\\6&1&0&0\\7&3&1&0\\3&5&9&1\end{bmatrix}\; lower unit triangular matrix

\begin{bmatrix}1&6&7&8\\0&1&7&3\\0&0&1&6\\0&0&0&1\end{bmatrix}\; upper unit triangular matrix

### Strictly Triangular Matrix:

In any upper or lower triangular matrix if the elements of the main diagonal are zero then it is called strictly upper or lower triangular matrix

\begin{bmatrix}0&6&7&8\\0&0&7&3\\0&0&0&6\\0&0&0&0\end{bmatrix}\; upper strictly triangular matrix.

\begin{bmatrix}0&0&0&0\\5&0&0&0\\6&1&0&0\\9&7&5&0\end{bmatrix}\; lower strictly triangular matrix.

## What is Inverse Matrix?

Let A be a square matrix of order n. A matrix B of order n is said to be the inverse of A if AB=BA=I_n

##### Note:

The inverse of an arbitrary square matrix may not exist.

### Example of Inverse Matrix ?

The Matrix A=\;\begin{bmatrix}1&-1\\-1&1\end{bmatrix} has no inverse.

An inverse of a matrix, if it exists, is unique. For suppose that A has two inverses, say B and C. Then

AB = BA=I and AC = CA= I

Therefore, by the associative law for multiplication, we have

B(AC) = BI = B

and (BA)C = IC = C

and so B = C

The unique inverse of a matrix A, if it exists, is denoted by A^{-1}

A square matrix A, whose inverse exists, is called a nonsingular or invertible matrix. Square matrices which do not have inverse are called singular matrices.

## Properties of Inverse Matrix Proof

It is easy to verify the following properties of inverses of matrices:

• For any invertible matrix A,(A^{-1})^{-1} = A
• For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}
• For any invertible matrix A, (A^t)^{-1} = (A^{-1})^t

### Inverse MatrixProof

For nonsingular matrices A and B, (AB)^{-1} = B^{-1} A^{-1}.

Since A and B are nonsingular, A^{-1} and B^{-1} exist. Also, Since A and B square matrices AB are defined.

To prove that AB is nonsingular and B^{-1} A^{-1} is the inverse of AB, We show that

(AB)

## What is Diagonal Matrix?

In mathematics, the term diagonals matrix define as the matrix in which the off diagonals entries are zero and main diagonals entries are some else. There is no restriction for main diagonals entries. diagonals matrix is also called the scaling matrix because when we multiply any matrix with diagonals matrix it change the scale of that matrix here the scale meaning size of the matrix. The identity matrix is one of the examples of the diagonals matrix. because in the identity matrix the main diagonals are 1 and off diagonals are zero.

### Diagonal Matrix Definition?

Diagonals matrix must be a square matrix. Since the square matrix is a symmetric matrix so we also called this matrix the symmetric diagonals matrix.

### Examples of diagonals matrix

##### Example no 1:
\begin{vmatrix}6&0\\0&4\end{vmatrix}

It is a 2 by 2 diagonals matrix.

##### Example no 2:
\;\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}

It is 3 by 3 diagonals matrix.

##### Example no 3:
\begin{bmatrix}7&0&0&0\\0&5&0&0\\0&0&4&0\\0&0&0&9\end{bmatrix}

it is a 4 by 4 diagonals matrix.

### Properties of Diagonals matrix

• When we add two diagonals matrices then the resultant matrix also diagonals matrix. \;\begin{bmatrix}1&0&0\\0&4&0\\0&0&3\end{bmatrix}\;\;+\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&9&0\\0&0&5\end{bmatrix}
• If any matrix is triangular and normal then then it is diagonal matrix.
• both the upper and lower triangular matrix are diagonals matrix.
• Symmetric matrix also diagonals matrix.
• When we multiply any matric with diagonals matric then the matrix which we obtain is diagonals matrix \;\begin{bmatrix}2&0&0\\0&4&0\\0&0&5\end{bmatrix}\;\;\times\;\;\begin{bmatrix}4&0&0\\0&3&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&12&0\\0&0&10\end{bmatrix}
• Zero matrix is diagonals matrix. \;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\; it is a 3 by 3 zero matrix and it hold the property of diagonal matrix.
• Identity matrix also diagonals matrix \;\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} this is the 3 by 3 identity matrix since it is hold the property of diagonals matrix
• The matrix of order 1 by 1 is diagonals matrix.
• Transpose of any diagonals matrix is same as matrix.
• Inverse of diagonals matrix is resiprocal of main diagonals entries .
• in under multiplication the diagonals matrix are commutative as AB = BA

### Note Diagonal Matrix

• since 1 by 1 order matrix is also diagonal matrix.
• there is no restrication for diagonal entries.

## what is Rank of Matrix?

The rank of Matrix A is equal to the number of nonzero rows in its echelon form.

or

The order of Ir in the canonical form of A.

To find the rank of any matrix we just reduce this matrix to its echelon form or canonical form and count its nonzero rows. p(A) shows the rank of matrix A.p(A) is the notation of the rank of a matrix. If a matrix has all elements are zero its rank is zero. If a matrix has not had full rank then it is said to be rank-deficient. The column rank and row rank are always equal. Moving towards firstly we know about the echelon form and canonical form.

A Matrix is in echelon form if it satisfies the following.

The first nonzero element in each row is 1.

Each leading entry is in a column to the right of the leading entry in the previous row.

A row echelon form is a canonical form. now we move to our main topic.

Example of the rank of a matrix

A=\begin{array}{c}\begin{array}{c}\begin{bmatrix}5&6&3\\1&0&8\\0&0&0\end{bmatrix}\end{array}\end{array}

### Properties of the Rank of Matrix

1. When the rank of a matrix is equal to its dimensions then it is said to be a full rank matrix.
2. When the rank of the matrix is less than its original matrix then the answer is in the form of inconsistent (in variables form).
3. If A has a square matrix then it is veritable if and only if it has full rank.
4. The rank is at least one
5. Only a zero matrix has zero ranks.
6. The rank cannot be larger than the smallest dimension of the matrix.

#### How we find the rank of a matrix

We take a square matrix of order 3 by 3

\begin{array}{c}\begin{array}{c}\begin{bmatrix}5&9&3\\-3&5&6\\-1&-5&-3\end{bmatrix}\end{array}\end{array}

to find rank of matrix we reduce this matrix in an echelon form.

\begin{bmatrix}-1&-5&-3\\-3&5&6\\5&9&3\end{bmatrix}\; by R_{13}

\begin{bmatrix}1&5&3\\-3&5&6\\5&9&3\end{bmatrix}\; by(-1) R_1

\begin{bmatrix}1&5&3\\0&20&15\\0&-16&-12\end{bmatrix}\; by R_2 +3 R_1 and R_3 -5 R_1

\begin{bmatrix}1&5&3\\0&1&3/4\\0&-16&-12\end{bmatrix}\; by\frac1{20} R_2

\begin{bmatrix}1&5&3\\0&1&3/4\\0&0&0\end{bmatrix}\; by R_3 +16 R_2

this is the echelon form

at last, we count the number of non zero rows that is rank of matrix here the non zero rows are 2 so the rank of thus matrix is 2.

### Why we Find the Rank of matrix?

The rank of the matrix tells us a lot about the matrix.

If we want to solve the system of linear equations we use the rank for this purpose. If the rank of any matrix equals the number of variables then we are able to find a unique solution of this matrix.

###### Note:
1. When we reduce a matrix into its echelon form we can only elementary row transformations.
2. the Nullity of a matrix n − r. where n = order of a matrix and r = rank of a matrix.
3. rank of a matrix is zero only it case when the matrix has all entries are zero.

## What is Row Matrix?

A Matrix, which has only one row is called Row Matrix. i.e.,1×n matrix of the form a_ia_{i_2}a_{i_3}...a_{in} is said to be a row matrix. A row matrix is also called row vector.

### Example of Row Matrix

A=\begin{bmatrix}1&2&1&0\end{bmatrix}
The order of the above matrix is 1 × 4 because it contain 1 Row and 4 Columns.

It is most common example of row matrix because it contains only one row. There is no limit for row matrix that it could contain only few elements or members.

.i.e., A=\begin{bmatrix}1&2&0&1&…&n\end{bmatrix}

it is most satisfying example to prove that a Row Matrix could contain more elements. Its order is 1× n. Because it contain 1 Row and n Columns.

### Common Examples of Row Matrix

Some common examples of Row Matrix are as follows;

• A=\begin{bmatrix}1&2\end{bmatrix}
• B=\begin{bmatrix}1&2&3\end{bmatrix}
• C=\begin{bmatrix}1&0&2&0\end{bmatrix}

#### Transpose of Row Matrix

Transpose of a row matrix is obtained by interchanging rows into columns and columns into rows.

#### Rank of a Row Matrix

Let A be a non zero matric. If r is the number of non zero rows then it is reduced to reduced echelon form , then r us called the (row) rank of the matrix A. In simple words Number of non zero rows in a matrix is called Rank. It is obtained by applying row operation.

## what is Column Matrix?

A Matrix which has only one column is called Column matrix. , i.e., m×1 matrix of the form
A=\begin{bmatrix}1\\2\\3\end{bmatrix} is said to be a Column Matrix or column vector.

#### Examples of Column Matrix

• A=\begin{bmatrix}1\\2\end{bmatrix}
• B=\begin{bmatrix}1\\2\\0\end{bmatrix}
• C=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}

#### Transpose of Column Matrix

Transpose of columns matrix is also as similar as Row matrix. It is also obtained by changing the order m×n into n×m.

#### Difference between Row Matrix and Column Matrix

Main difference between Row and Column matrix it that Row matrix contain single row and Column matrix contain single column.

#### Addition of Matrix

Addition of a matrix is only possible when order of both matrix is equal. It means that number of rows and number of columns should be equal. It is obtained by adding each entry of A to corresponding entry of B.

##### Row and column matrix in the case of Determinant

(i) If two rows in a matrix are identical or two columns are identical(means if these are totally same), their determinant would be zero. It means \left|A\right|=0.

##### “Elementary Row and Column Operation of Matrix“

Usually a given system of linear equation is reduced to a simple equivalent system by applying in turn finite number of elementary operations which are:

• Interchanging two equations
• Multiplying an equation by a non zero number
• Adding a multiple of one equation to another equation.
###### Corresponding to these three elementary operations, the following elementary operations are applied to matrices to obtain equivalent matrices:
• Interchanging two rows.
• Multiplying a row by a non zero number.
• Adding a multiple of one row to another row.

#### EXAMPLE OF ROW OPERATION IS:R2 –> R2

Now we state that elementary column operations are:

• Interchanging two columns.
• Multiplying a column by a non zero number.
• Adding a multiple of one column to another column.

Marices A and B are equivalent if B can be obtained by applying in turn a finite number of row operations on A.

## What is skew Symmetric

Let A be a square matrix if the transpose of this matrix is equal to the negative of that matrix then it is called skew symmetric matrix. In linear algebra skew-symmetric matrix defines as the transpose of the matrix that is equal to its negative matrix. The skew-symmetric matrix must be a square matrix. In the skew-symmetric matrix, the main diagonals entries are zero.

We also define skew-symmetric matrix as A^t = -A. Let A be a square matrix of order n by n  we denote the entries as aij in skew-symmetric when we take the transpose of matrix A then these entries will be changed (aij)^t = -(aij) where i denote the number of rows and j shows the number of columns.

### Example of skew-symmetric matrix

No 1.

Skew-symmetric matrix of A:

A=\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix},

Order of this matrix is 3 by 3.

A^t=\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix}^t \Rightarrow A^t=\begin{bmatrix}0&2&-3\\-2&0&-4\\3&4&0\end{bmatrix} \Rightarrow A^t=-\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix}

A^t= –A

hence this is skew symmetric matrix.

No 2.

skew-symmetric matrix of B:

B=\begin{bmatrix}0&-2\\2&0\end{bmatrix}

Order of this matrix is 2 by 2.

B^t=\begin{bmatrix}0&-2\\2&0\end{bmatrix}^t \Rightarrow B^t=\begin{bmatrix}0&2\\-2&0\end{bmatrix} \Rightarrow B^t=-\begin{bmatrix}0&-2\\2&0\end{bmatrix}

B^t= –B

so this matrix is skew symmetric matrix.

#### Properties of a skew-symmetric matrix:

• When we have scalar product of two skew symmetric matrices then the resultant matrix also skew symmetric matrix.
• Eigen value of real skew symmetric matrices are zero however non zero symmetric matrices have non zero eigen values.
• when we add two skew symmetric matrices then the matrix that we obtain is skew symmetric matrix.
• the trace of skew symmetric matrices are zero because the trace is the sum of main diagonals and the main diagonals of the skew symmetric matrices are zero.
• the determinant of skew symmetric matrices are having odd order that is zero.
• when we added identity matrix in any skew symmetric matrix then the matrix that we obtain is invertible.
• skew symmetric matrix is diagonalizable.
• The rank of skew symmetric matrices are always even.
Here the question is arises that why the diagonals elements of skew symmetric matrix are always zero? The diagonals elements of skew symmetric matrix are always zero because when the main diagonals are non zero then the transpose and negative of that matrix does not equals. let we explain this in detail.

Let A be a square matrix

A=\begin{bmatrix}1&2\\-2&4\end{bmatrix}

first, we take the transpose of this matrix. for transpose the first-row change in the first column and second-row change in the second column.

A^t=\begin{bmatrix}1&-2\\2&4\end{bmatrix}

Now we take negative of this matrix for this we change the negative value into positive and positive values in negative.

-A=\begin{bmatrix}-1&-2\\2&-4\end{bmatrix}

A^t not is equal to A

here the property of skew-symmetric does not hold two matrices are equal if and only if when their order is same and all elements in matrices are same and here both matrices are different so the property of skew-symmetric that the transpose of a matrix is equal to its negative matrix is not hold

##### Note skew-symmetric matrix

since the skew-symmetric matrix is singular therefore the inverse of the skew-symmetric matrix does not exist.

## Define Symmetric Matrix with Examples

In linear algebra, A square matrix A is called a symmetric matrix if we take a transpose of a matrix and the answer is itself matrix.

such that Symmetric Matrix

### Symmetric Matrix Example 3×3

A^t = A

A symmetric matrix is always a square matrix because equal matrices have equal dimension. In the symmetric matrix, the number of rows and the number of columns are the same as it is a square matrix. Every real symmetric matrix can be diagonalizable.

### General form of Symmetric matrix

A symmetric matrix is important and useful in many applications because of its application. Some examples of well-known symmetric matrices are the correlation matrix covariance matrix and distance matrix.

Before we move further first we discuss that what is a square matrix is and what is the transpose of a matrix

A matrix is said to be a square matrix if and only if the number of columns of rows is equal to the number of columns. A number of rows shown by m and number of columns are shown by n. in square matrix m = n.

### Transpose of a symmetric matrix

If we want to take the transpose of any matrix then we interchanging the rows and columns of the original matrix. If a matrix A has m × n order and we take transpose then the order is changing and the order is n × m.

### Example of symmetric matrix

A=\;\begin{bmatrix}1&2\\2&1\end{bmatrix}

order of this matrix A is 2 by 2

Firstly we change the first row to the first column

Then we change the second row to the second column

### Example of a symmetric matrices

A=\;\begin{bmatrix}3&-2&4\\-2&6&2\\4&2&3\end{bmatrix}

Here the order of the matrix A is 3 -by 3.

### Properties of symmetric matrices

• The product of two symmetric matrices may not be symmetric. That is if

A^t = A , B^t = B then (AB)^t = B^t A^t = BA not equal to AB is general.it is only possible when A and B are commute like AB = BA

• The sum of two symmetric matrices is again a symmetric matrices.
• The difference of two symmetric matrices is also again a symmetric matrice.
• The most important property of symmetric matrices is that their eigenvalues behave very nicely.
• Hermitian matrice is also symmetric matrix if the entries are complex number.
• If all eigenvalues of symmetric matrix A are different  then the  matrix A can transformed into its diagonal matrice.
• There is no complex number in Eigen value of a symmetric matrix that is means there are all real numbers.
• Symmetric matrices has linearly independent eigen vector.

#### Note symmetric matrix

The zero matrix is also a symmetric matrix.

## what is Identity matrix?

An Identity matrix is a matrix in which the diagonals entries are 1 and off diagonals entries are zero. An Identity matrix is always a square matrix.

As we know that square matrix is in which a number of rows is equal to the number of columns. The identity matrix is denoted by I_{n\times n} where n by n shows the order of the identity matrix.it plays an important role in linear algebra since.

• If we multiply any real number with 1 then its remains unchanged.
• If we multiply any matrix with identity matrix then its remains unchanged.

## General form of identity matrix

#### Identity Matrix Examples

Single matrix

I =\begin{bmatrix}1\end{bmatrix}

The order of the above matrix is 2 by 2

i_2 =\begin{bmatrix}1&0\\0&1\end{bmatrix}

The order of this matrix is 4 by 4

i_4 =\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}

The order of the above matrix is n by n

i_n =\begin{bmatrix}1&0&0&…&0\\0&1&0&…&0\\0&0&1&…&0\\…&…&…&…&…\\0&0&0&…&1\end{bmatrix}

### Notations of Identity Matrix

The term unit matrix is widely used at an early age but now the term identity matrix has become a standard form. There is more notation used in different books such that in group theory the identity matrix is sometimes denoted by boldface one like 1. Some mathematics books use U or E where U stands for unit matrix and E stand for a germen word Einheitsmatrix.

There is one more notation for identity matrix is KRONECKER DELTA.

If we have a matrix A with order m by n then it is a property of multiplication is that

l_mA= AI_n= A.

### properties of identity matrix

1. If we multiplying the two inverse matrices then we always get identity matrix in result such that
2. If we multiply any matrix with identity matrix the result is identity matrix

I_mC = C = CI_n

• The identity matrix is the only matrix which is idempotent matrix with non-zero determinant that means it is the only matrix which have
• When the multiply by itself the result Is itself
• All of rows and columns are linearly independent
• The square root of  an identity matrix is itself and this is the only positive definite square root
• It is always square matrix as the number of rows and the number of columns are equal