Diagonalization

Diagonalization

Lessons

An n×nn \times n matrix AA is diagonalizable if and only if AA has nn linearly independent eigenvectors.

If you have nn linearly independent eigenvectors, then you can use the formula

A=PDP1A=PDP^{-1}

where:
The columns of PP are the eigenvectors
The diagonal entries of DD are eigenvalues corresponding to the eigenvectors
P1P^{-1} is the inverse of PP

To see if a matrix is diagonalizable, you need to verify two things
1. There are n linearly independent eigenvectors
2. AP=PDAP=PD

Useful fact: If AA is an n×nn \times n matrix with nn distinct eigenvalues, then it is diagonalizable.
  • 1.
    Diagonalization Overview:
    a)
    The Formula A=PDP1A=PDP^{-1}
    • Why is it useful?
    • Finding High Powers of AA

    b)
    How to Diagonalize a Matrix
    • Calculate the eigenvalue
    • Find the eigenvectors
    • Combine the eigenvectors to create PP
    • Use the eigenvalues to create DD
    • Find P1P^{-1}

    c)
    How to See if a Matrix is Diagonalizable
    • Finding the basis of each eigenspace
    • Create a Matrix PP and Matrix DD
    • Check if AP=PDAP=PD


  • 2.
    Computing a Matrix of High Power
    Let A=PDP1A=PDP^{-1}, then compute A4A^4 if
    Computing a Matrix of High Power

  • 3.
    Determining if a Matrix is Diagonalizable
    Is the following matrix diagonalizable?
    Is this matrix diagonalizable

  • 4.
    Is the following matrix diagonalizable?
    Is this matrix diagonalizable

  • 5.
    Diagonalizing the Matrix
    Diagonalize the following matrix.
    Diagonalize this matrix

  • 6.
    Proof relating to Diagonalization
    Show that if A is diagonalizable and invertible, then so is A1A^{-1}.