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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

What is a matrix and why do we need matrix?

By Kardi Teknomo, PhD.
LinearAlgebra

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When you have a numerical data, you may want to put your data into a table.
For example: the unit price of land, labor and material in 4 cities is given in the following table
What is matrix

When you give name to your data table which consists of horizontal rows and vertical columns and think of it as one entity, you have a compact form that make it easier to manipulate your data and to automate the operations on your data using fast and efficient procedures to find the solution of your problem.

Since the meaning of each row and each column is well defined in each application, we can remove the definition and dealing only with the array of elements. The table above can be written as
What is matrix

A collection of numerical data which is organized into rows and columns is called a matrix. A matrix is rectangular array of elements. A matrix is defined by its size that is the number of rows and the number of columns. We always write matrix size as number of rows first and then the number of columns. A square matrix happens when the number of rows is equal to the number of columns, the size can be written as one number called order of the matrix, that is equal to the number of rows = the number of columns.

Example:
A (m by n) matrix has m rows and n columns.
When m=2 and n = 3, we have for instance
What is matrix

Example:
A (n by n) matrix is a square matrix of order n. When n = 3, we have for instance
What is matrix

Based on the matrix size, we can give name:
·         A rectangular matrix is general matrix size m by n
·         A square matrix order n has equal size n by n (the number of rows is the same as the number of columns)
·         A vector is a matrix size m by 1 (any positive number of rows, but only one column)
·         A row vector is a matrix size 1 by n (any positive number of columns, but only one row)

In many modern books and journal papers, a matrix usually has notation of a bold uppercase roman letter, such asWhat is matrix,What is matrix,What is matrix, …, What is matrixwhile a vector is written as bold lowercase roman letter such asWhat is matrix,What is matrix,What is matrix, …, What is matrix. The entry element of a matrix or a vector is a scalar, usually denoted as lowercase roman letter.


For example
What is matrix

The following are some examples of useful ways to organize things into matrix:

  • One of the most useful matrix applications is to represent simultaneous linear equations. Consider a linear system
    What is matrix
    It can written as an augmented matrix that can be solved using Gauss Jordan method
    What is matrix
  • In some data matrix, each row in a matrix usually represents an object or one observation, while each column of a matrix usually indicates features (or attributes or variables).
    What is matrix
  • In distance matrix, each row and each column represents the same object, the element of the matrix indicate the distance between objects. The distance matrix is a symmetric matrix.
    What is matrix
  • A matrix can also represents observations in rows and other observations in columns, or some attributes in rows and some other attributes in columns.
    What is matrix
  • Some types of matrices are closely related to graph theory. The rows or columns can represent the nodes or links in a graph. For example, the incidence matrix below is associated with the 3 nodes and 4 links graph.
    What is matrixWhat is matrix
  • Probability matrix contains probability to move from one position to the next position.

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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