Mathematical Formulation Consider a transformation matrix $latex {A}&s=1$, such that it transforms a vector $latex {X}&s=1$ into $latex {Y}&s=1$. Thus, we can write, $latex {Y = AX}&s=2$ Recall that 2 vectors directed along same direction are simply scalar multiples of each other. e.g. $latex {\vec P = 3 \hat i + 4 \hat j =… Continue reading Eigenvalues and Eigenvectors of Matrices

# Category: M I

## Matrices and Simultaneous Equations

Prerequisite : How To Find Rank of a Matrix? System of Linear Algebraic Equations Consider the following equations: $latex {2x+3y =8, \ x - y = -1}&s=2$ This can be written using matrix form as $latex {\begin {bmatrix} 2 & 3 \\ 1 & -1 \end {bmatrix} \begin {bmatrix} x \\ y \end {bmatrix} =… Continue reading Matrices and Simultaneous Equations

## Rank of a Matrix

Rank of a Matrix An important characteristic of any matrix is its rank. It tells us the number of independent rows or columns of matrix. Note that all matrices have a rank, unlike inverse of a matrix, which only a non-singular square matrix has. Definition: A matrix has a rank $latex {r}&s=1$, if I) At… Continue reading Rank of a Matrix

## Errors and Approximations, Maxima and Minima, Lagrange’s Method

Errors and Approximations If $latex {f}&s=1$ is a function of $latex {x,y,z}&s=1$, then the error in $latex {f}&s=1$ is $latex {df = \frac {\partial f}{\partial x} dx + \frac {\partial f}{\partial y} dy + \frac {\partial f}{\partial z} dz}&s=2$ However, this is an approximation. Note: If necessary, we take log on both sides. Maxima and… Continue reading Errors and Approximations, Maxima and Minima, Lagrange’s Method

## Jacobians

Introduction To get an idea about Jacobians, one requires knowledge of matrices, determinants, functions and partial differentiation. I) Matrices and Determinants : A matrix is a rectangular arrangement of numbers in $latex {m}&s=1$ rows and $latex {n}&s=1$ columns. When $latex {m =n}&s=1$, the matrix is known as a square matrix. Determinants are defined for square… Continue reading Jacobians

## Partial Differential Equations

Introduction So far, while studying calculus, we have dealt with functions of single variable, i.e. $latex {y=f(x)}&s=1$. $latex {sin (x^2), ln \ x, e^{cos \ (tan \ x)}}&s=1$ are few examples. Irrespective of their complexity, the variable $latex {y}&s=1$ always depended on the value of independent variable $latex {x}&s=1$. We also defined the derivatives and… Continue reading Partial Differential Equations

## Infinite Series

Sequence and Series are fundamentally different from each other. Sequence A sequence is an arrangement of numbers (or objects). For example, $latex {3, 7,11,15,19, \cdots}&s=2$ is a sequence. The first term is $latex {a_1 = 3}&s=1$, the second term is $latex {a_2 = 7}&s=1$. In general, $latex {n}&s=1$th term in a sequence is denoted by… Continue reading Infinite Series