What are Matrices?
At a very high level, you can think of a Matrix (Plural : matrices) as two dimensional array of numbers, symbols or expressions. For example – following image depicts a MxN matrix, where it has M-rows and N-Columns. A specific element is being denoted by Ai,j, which means the element value at the ith row and jth column.
- Size of matrix: The size of a matrix is the number of columns and the rows in a given matrix (M x N).
- Row Vector: matrices with the single row are called row vectors (1 x N)
- Column Vector: matrices with single column are called column vectors (M x 1)
- Square Matrix: a matrix with the same number of rows and columns is called square matrix (N x N)
- Infinite Matrix: a matrix with infinite number of rows or columns is called infinite matrix
- Empty Matrix: a matrix with zero number of rows and columns is called an empty matrix
- Identity Matrix: the identity matrix is a N x N matrix with 1 at the diagonal and 0 else where
- Inverse Matrix: A matrix is an inverse matrix of a given matrix, if their multiplications results into identity matrix
Read the reference links to know the details about matrix operations. Here is the summary of the commonly used operations:
- Adding/subtracting matrices
- two matrices of exactly same dimensions can be added or subtracted element by element
- Scalar Multiplication
- Multiplying a matrix by constant number (negative or positive) leads to multiplication of all the element by that specific number.
- Matrix Multiplications
- two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second
- You can calculate square of a matrix only when they have same number of rows and columns (i.e. they are square matrix)
- The transpose of a matrix results into a MxN matrix becoming a NxM matrix. Here the rows become columns and the columns becomes rows
Usage of matrices
As you know that matrices are the tabular representation of data, essentially anything which requires two dimensional numerical data representation, can make use of the matrices.
Following is the list of more obvious usage of the matrices
- Demographic representation of data for a specific topic
- Projection of 3-dimensional images into two dimensional screens
- Website’s page ranking
- Plotting graphs
- In Seismic Surveys, to produce detailed images of local geology to determine the location and size of possible oil and gas reservoirs
- In robotics & automation in terms of base elements for the robot movements, including rotations and translations through planes
- It is used in cryptography to encrypt / decrypt messages
- Matrices allow complex dutching/betting combinations without separate formulae such as multiple complex simultaneous equations
- In the gaming, the uses of matrices allows effective algorithms around collision and ray tracing
- Markov Chain makes use of matrices extensively, which in turn is used to solve various real life problems including calculation of possibilities (as in trades or prediction of some events).