Mathematical Fundamentals: Linear Algebra Jothi

Transcript:Mathematical Fundamentals: Linear Algebra Jothi:
Mathematical Fundamentals: Linear Algebra Jothi Ramalingam jothiram@nitk.edu.in Introduction: Diagonal Matrices Before beginning this topic, we must first clarify the definition of a “Diagonal Matrix”. A Diagonal Matrix is an n by n Matrix whose non-diagonal entries have all the value zero. Introduction: Diagonal Matrices In this presentation, all Diagonal Matrices will be denoted as: where dnn is the entry at the n-th row and the n-th column of the Diagonal Matrix. Introduction: Diagonal Matrices For example, the previously given Matrix of: Can be written in the form: diag(5, 4, 1, 9) Introduction: Diagonal Matrices The Effects of a Diagonal Matrix The Identity Matrix is an example of a Diagonal Matrix which has the effect of maintaining the properties of a Vector within a given System. For example: Introduction: Diagonal Matrices The Effects of a Diagonal Matrix However, any other Diagonal Matrix will have the effect of enlarging a Vector in given axes. For example, the following Diagonal Matrix: Has the effect of stretching a Vector by a Scale Factor of 2 in the x-Axis, 3 in the z-Axis and reflecting the Vector in the y-Axis. The Goal By the end of this PowerPoint, we should be able to understand and apply the idea of Diagonalisation, using Eigenvalues and Eigenvectors. The Matrix Point of View By the end, we should be able to understand how, given an n by n Matrix, A, we can say that A is Diagonalisable if and only if there is a Matrix, δ, that allows the following Matrix to be Diagonal: And why this knowledge is significant. The Goal The Square Matrix, A, may be seen as a Linear Operator, F, defined by: Where X is a Column Vector. Linear : A(x+y) = Ax + Ay The Points of View Furthermore: Represents the Linear Operator, F, relative to the Basis, or Coordinate System, S, whose Elements are the Columns of δ. The Points of View If we are given A, an n by n Matrix of any kind, then it is possible to interpret it as a Linear Transformation in a given Coordinate System of n-Dimensions. For example: Has the effect of 45 degree Anticlockwise Rotation, in this case, on the Identity Matrix. The Effects of a Coordinate System However, it is theorised that it is possible to represent this Linear Transformation as a Diagonal Matrix within another, different Coordinate System. We define the effect upon