Advanced Matrix Operations – A theoretical view

                     

  Advanced Matrix Operations – A theoretical view               ========================================

 

* One may encounter some important matrix operations using algorithmic formulations

 

* The advanced matrix operations are formulating the transpose and inverse of any given matrix form of dataset

 

* Transposition occurs when a matrix of shape n x m is transformed into a matrix in the form of m x n by exchanging the rows with the columns

 

* Most of the tests indicate the operation using the superscript T in the form of A( transpose )

 

* One can apply " matrix inversion " over matrices of shape m x m , which are square matrices that have the same number of rows and columns . In mathematical language , this form of square ordering of matrices is said that the matrix has m rows and m columns .

 

* The above operation is important for the sake of finding the immediate resolution of the various equations which involve matrix multiplication such as y = bX where one has to discover the values in the vector b . More on Matrix multiplications with more conceptual examples would be showcased in another article in which I shall try to cover how the Matrix Multiplication of different Matrices occur and how this Multiplication is used to solve more important / complex problems .

 

 

* Since most scalar numbers (exceptions including zero) have a number whose multiplication results in a value of 1 , the idea is to find a matrix inverse whose multiplication would result in a special matrix called the identity matrix whose elements are zero , except the diagonal elements

 ( the elements in positions where the index 1 is equal to the index j)

* Now , if one wants to find the inverse of a scalar quantity , then one can do so by finding the inverse of a scalar . (The scalar number n has an inverse value that is n to the power minus 1 which can be represented by 1/n that is 1 upon n )

 

* Sometimes, finding the inverse of a matrix is impossible and hence the inverse of a matrix A is indicated as A to the power minus 1

 

* When a matrix cannot be inverted, it is referred to "singular matrix" or a "degenerate matrix" . Singular matrices are usually not found in isolation, rather are quite rare to occur and generalise .