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### Linear Algebra - Matrices Part I - A Tutorial with Examples ------------xxxx------------

Linear Algebra: Matrices Part 1

## Linear Algebra - Matrices Part I - Outline of Contents:

 Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular  matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.

Matrices

This tutorial contains an introduction to various terms and definitions used while dealing with Matrices

A Quick Introduction (Covered in greater detail in the tutorial document at the end)

A rectangular arrangement of numbers (which may be real or complex numbers) in rows and columns, is called a matrix. This arrangement is enclosed by small ( ) or big [ ] brackets. The numbers are called the elements of the matrix or entries in the matrix. Example:-

1 2 3
4 5 6
7 8 9

A matrix having m rows and n columns is called a matrix of order m×n matrix (read as an m by n matrix). We will frequently use this notation A=[ ]m×n represents the element in the i-th row and the j-th column in a matrix of order m×n.

Two matrices A and B are said to be equal matrix if they are of same order and their
corresponding elements are equal. Example:

A =

1 2 3
4 5 6
7 8 9

B =
1 2 3
4 5 6
7 8 9

#### 1. Row matrix:

The matrix has only one row and any number of columns. Example

[ 1 2 3 4 ]

#### 2. Column matrix:

The matrix has only one columns and any number of rows. Example

1
2
3

#### 3. Singleton matrix:

If a matrix has only one element. Example 

#### 4. Null or Zero matrix:

All the elements are zero in such a matrix. Example:

0 0 0
0 0 0
0 0 0

#### 5. Square matrix:

If the number of rows and columns in a matrix are equal, then it is
called a square matrix. Thus A=[ ] m×n is a square matrix if m=n

#### 6. Trace of a matrix:

The sum of the diagonal elements of a square matrix. A is called the trace
of matrix A, which is denoted by tr A=a11+a22+…….ann

#### 7. Diagonal Matrix:

All elements not on the principal diagonal are zero

Example:
5 0 0
0 1 0
0 0 4

#### 8. Identical Matrix:

A diagonal matrix in which all elements on the principal diagonal are set to one.

#### 9. Scalar Matrix:

A diagonal matrix in which all elements on the principal diagonal are equal.

## Other concepts introduced in the tutorial :

Triangular Matrices: Upper and Lower triangular matrices
Addition, subtraction and scalar multiplication of matrices
Multiplication of matrices
How to compute the minors, cofactors, adjoint, transpose and inverse of a matrix. Properties of the adjoint,  inverse and transpose of a matrix
Special types of matrices: Symmetric matrix, skew-symmetric matrix, singular matrix, non-singular matrix, orthogonal matrix, idempotent matrix, involuntary matrix, nilpotent matrix, unitary matrix, periodic matrix, hermitian matrix, skew-hermitian matrix, conjugate of a matrix

## Tutorial :

Matrices: Brief summary of things we read in the above tutorial.

#### Definition:

A rectangular arrangement of numbers (which may be real or complex numbers) in rows and columns, is called a matrix. This arrangement is enclosed by small ( ) or big [ ] brackets. The numbers are called the elements of the matrix or entries in the matrix.

#### Order of a matrix:

A matrix having m rows and n columns is called a matrix of order m×n matrix (read as an m by n matrix).

#### Equality of Matrices

Two matrices A and B are said to be equal matrix if they are of same order and their corresponding elements are equal.

#### Types of Matrices:

1. Row matrix: The matrix has only one row and any number of columns.
2. Column matrix: The matrix has only one columns and any number of rows.

3. Singleton matrix: If a matrix has only one element.

4. Null or Zero matrix: All the elements are zero in such a matrix.

5. Square matrix: If the number of rows and columns in a matrix are equal, then it is called a square matrix.

#### Trace of a matrix

The sum of the diagonal elements of a square matrix. A is called the trace of matrix A

#### Diagonal Matrix

If all elements except the principal diagonal in a square matrix are zero, it is called a diagonal matrix.

#### Identity Matrix

A square matrix in which elements in the main diagonal are all 1 and rest are all zero

#### Scalar matrix:

A square matrix whose all non-diagonal elements are zero and diagonal elements are equal is called a scalar matrix.

#### Triangular Matrix

A square matrix [ ] is said to be triangular matrix if each element above or below the principal diagonal is zero

We will add the corresponding elements.

#### Subtraction of matrices:

We will subtract the corresponding elements.

#### Scalar multiplication of matrices

We will multiply the corresponding elements with the given scalar. (Explained in the tutorial above)

#### Multiplication of matrices

Two matrices A and B are conformable for the product AB if the number of columns in A (pre multiplier) is same as the number of rows in B(post multiplier).Thus, if A=[ ]m×n and B=[ ]n×p are two matrices of order m×n and n×p respectively then there product is of order m×p (Explained with examples in the tutorial above).

#### Recap of the basic properties of matrix addition

If A,B,C are three matrices such that the product is defined, then

(i) AB≠BA

(ii) (AB)C=A(BC)

(iii)IA=A=AI

(iv) A(B+C)=AB+AC

(v) If AB=AC

(vi) If AB=0

#### Transpose of a Matrix:

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of Matrix A and is denoted by AT

Special Types of Matrices:

1. Symmetric Matrix
2. Skew-Symmetric Matrix
3. Singular Matrix:
4. Non Singular Matrix:
5. Orthogonal Matrix:
6. Idempotent matrix:
7. Involuntary matrix:
8. Nilpotent matrix:
9. Unitary matrix:
10. Periodic matrix: A matrix is called a periodic matrix if Ak+1=A where k is a positive

integer. If however k is the least positive integer for which Ak+1=A then k is said to

be the period of A

11. Hermitian matrix

12. Skew Hermitian matrix

13. Conjugate of matrix: The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding
conjugate complex numbers is called conjugate of A.

14. Submatrix

15. Transpose of a conjugate of a matrix

### You might like to take a look at some of our other Linear Algebra tutorials :

 Introduction to Matrices - Part I   Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices. Introduction to Matrices - Part II Problems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form. Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants. Simultaneous linear equations in multiple variablesRepresenting a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. Basic concepts in Linear Algebra and Vector spacesTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. Introductory problems related to Vector Spaces - Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a sub-space, demonstrating that something is not a sub-space of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices. More concepts related to Vector SpacesDefining and explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these. Problems related to linear transformation, linear maps and operators - Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem Eigenvalues, eigenvectors, Cayley Hamilton Theorem More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors  Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . A few closing problems in Linear AlgebraSolving a recurrence relation, some more of system of equations.