Linear Algebra - Matrices & Vectors

A matrix is a rectangular array of numbers or functions which will be enclosed in brackets. Matrix has rows, horizontal lines of entries and columns, vertical lines of entries. A matrix of any size m x n is called a rectangular matrix.

Square matrix, has as many rows as columns. Matrices of different sizes cannot be added. Matrix addition is commutative and associative.

Matrix having just a single row or column is called vectors. Row vector has one row and Column vector has one Column.

Matrix Multiplication

Transposition

The transpose of a matrix is obtained by writing its rows as columns (or equivalently its columns as rows). Transposition converts row vectors to column vectors and vice versa.

Rules of Transposition

Symmetric Matrix

Square matrix whose transpose equals to the matrix itself.

Skew Symmetric Matrix

Square matrix whose transpose equals to the minus of matrix.

Upper Triangle Matrix

Square matrix having non zero entries on and above the main diagonal.

Lower Triangle Matrix

Square matrix having non zero entries on and below the main diagonal.

Diagonal Matrix

Square matrix having non zero entries only on the main diagonal.

Unit Matrix

Diagonal matrix having all its main diagonal element are 1.

Stochastic Matrix

Square matrix with all entries non negative and all columns sums equal to 1.

Matrices are used to solve systems of linear equations. The approach to solving linear systems is called the Gauss elimination method.

The system is called linear because each variable x appears in the first power only, just as in the equation of a straight line.

x - variable ; a - coefficient ; b - numbers

If all the b are zero, then the system is called a homogeneous system. If at least one b is not zero, then the system is called a nonhomogeneous system.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Cache Memory

The supplementary memory system that temporarily stores frequently used instructions and data for quicker processing by CPU of a computer. Main memory consists of up to 2n addressable words, with each word having a unique n-bit address. there are M = 2n/K fixed length of blocks in main memory. where K - words The cache consists of m blocks called lines. The length of a line, not including tag and control bits, is the line size. As there are more blocks in main memory than lines in cache, an individual line cannot be uniquely and permanently dedicated to a particular block. Thus, each line includes a tag that identifies which particular block is currently being stored. The cache connects to the processor via data, control, and address lines. When a cache hit occurs, the data and address buffers are disabled and communication is only between processor and cache, with no system bus traffic. When a cache miss occurs, the desired address is loaded onto the system bus and the data are r...

Decision Tree - Gini Index

The Gini index is used in CART. The Gini index measures the impurity of the data set, where p i - probability that data in the data set, D belong to class, C i and pi = |C i,D |/|D| There are 2 v - 2 possible ways to form two partitions of the data set, D based on a binary split on a attribute. Each of the possible binary splits of the attribute is considered. The subset that gives the minimum Gini index is selected as the splitting subset for discrete valued attribute. The degree of Gini index varies between 0 and 1. The value 0 denotes that all elements belong to a certain class or if there exists only one class, and the value 1 denotes that the elements are randomly distributed across various classes. A Gini Index of 0.5 denotes equally distributed elements into some classes. The Gini index is biased toward multivalued attributes and has difficulty when the number of classes is large.

The Data Ilm

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