Diagonalization

Diagonalization

1. For the First Case

   • Matrix A is diagonalizable when A is similar to a diagonal matrix
     • A is diagonalizable when there exists an invertible matrix P such that is a diagonal matrix
   • Matrix B is said similar to matrix A if there is a nonsingular matrix P such that
     • If A and B are similar n x n matrices, then they have the same eigenvalues
     • Condition for diagonalization : A is diagonalizable iff it has n linearly independence eigenvectors
     • If the eigenvalues of an n x n matrix A are all distinct, then A is diagonalizable
   • Prodcedure for Diagonalizing An n x n Matrix A
     • Step 1: Find eigenvalue
     • Step 2: Form matrix P
     • Step 3: The matrix D = P^(-1)AP will be a diagonal with its principal diagonal entries are the eigenvalues corresponding to eigenvectors.
   • Symmetrix Matrices & Orthogonal Matrix
     • A square matrix A is symmetric when it is equal to its transpose
       • Properties of Symmetric Matrices
         • 1. A is diagonalizable
         • 2. All eigenvalue of A are real
         • 3. If lambda is an eigenvalue of A with multiplicity k, then lambda has k linearly independent eigenvectors. That is eigenspace of lambda has dimension k.
     • A nonsingular matrix A is called orthogonal matrix if A inverse = A transpose
       • The n x n matrix A is orthogonal iff the column (rows) of A form an orthonormal set of verctors in R^n
       • Let A be an n x n symmetric matrix . If lambda 1 and lambda 2 are distinct eigenvalues of A, then corresponding eigenvectors x1 and x2 are orthogonal
       • If A is a symmetric n x n matrix, then there exists an orthogonal matrix P such that (below) is a diagonal matrix

2. Eigenvector Problem

   • Diagonalization Problem: whether there exist or not an invertible matrix P such that (below) is a diagonal matrix?
     • If A is an n x n matrix, then A is diagonalizable and A has n linearly independent eigenvectors
     • Distinct eigenvalues of A -> Distinct eigenvectors of A -> then eigenvectors is a Linearly Independent set
     • If A is n x n diagonalizable square matrix and (below) is the diagonal matrix, then

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