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Linear Algebra - Eigenvalues, Eigenvectors and Cayley Hamilton Theorem

Linear Algebra


                                            Linear Algebra: Eigenvalues, Eigenvectors, Cayley-Hamilton Theorem

Linear Algebra - Eigenvalues, Eigenvectors and Cayley Hamilton Theorem - Outline of Contents:

 Eigenvalues, eigenvectors, Cayley Hamilton Theorem

Here's a quick outline of topics to be covered in this tutorial:


If all the minors of the matrix A of order r+1 are zero and at least one minor of order r≠0 then matrix A is said to be of rank ‘r’

Echelon form of Matrix

A matrix ’A’ is said to be in echelon form if,
1. All the zero rows of ‘A’ followed by non-zero row
2. The number of zero’s before the first non-zero element in a non-zero row is less than the number of such zeros in the next row
3. The first non-zero element in a non-zero row is ‘1’

Elementary Transformations or Elementary operations of a matrix -and what they result in:

The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations. We'll take a look at what happens after making these elementary transformations.
1. Interchange of any two rows(columns)
2. Multiplying all elements of a row(column) of a matrix by a non-zero scalar
3. Adding to the elements of a row (column) ,the corresponding elements of any other row(column) multiplied by any scalar k.

A major focus of this tutorial will be to introduce Eigenvalues and Eigenvectors

  • How to compute the Eigen-values and Eigen-Vectors of a Matrix.
  • Finding the characteristic equations of a Matrix

We'll also introduce important theorems such as :

Rank of the matrix ‘A’ is the number of non-zero rows in its echelon form
The product of the Eigen values of a square matrix A is |A|
Cayley Hamilton Theorem: Every square matrix of order n satisfies its own characteristic equation
... and so on

Complete Tutorial :

You might like to take a look at some of our other Linear Algebra tutorials :

 Introduction to Matrices - Part I   Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nil-potent, Singular, Non-Singular, Unitary matrices.

Introduction to Matrices - Part II Problems and solved examples based on the sub-topics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form.

Determinants Introduction to determinants. Second and third order determinants, minors and co-factors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants.  Simultaneous linear equations in multiple variablesRepresenting a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and non-homogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. 

Basic concepts in Linear Algebra and Vector spacesTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. Introductory problems related to Vector Spaces - Problems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a sub-space, demonstrating that something is not a sub-space of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices.

More concepts related to Vector SpacesDefining and explaining the norm of a vector, inner product, Graham-Schmidt process, co-ordinate vectors, linear transformation and its kernel. Introductory problems related to these.

Problems related to linear transformation, linear maps and operators - Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. 
Definitions of Rank, Eigen Values, Eigen Vectors, Cayley Hamilton Theorem 
Eigenvalues, eigenvectors, Cayley Hamilton Theorem

More Problems related to Simultaneous Equations; problems related to eigenvalues and eigenvectors  Demonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . 

 A few closing problems in Linear AlgebraSolving a recurrence relation, some more of system of equations.