Eigenvalues & Eigenvectors (DP IB Maths: AI HL)

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Naomi C

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Naomi C

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Characteristic Polynomials

Eigenvalues and eigenvectors are properties of square matrices and are used in a lot of real-life applications including geometrical transformations and probability scenarios. In order to find these eigenvalues and eigenvectors, the characteristic polynomial for a matrix must be found and solved.

What is a characteristic polynomial?

  • For a matrix bold italic A, if bold italic A bold italic x equals lambda bold italic xwhen bold italic x is a non-zero vector and lambda a constant, then lambda is an eigenvalue of the matrix bold italic A and bold italic x is its corresponding eigenvector
  • If bold italic A bold italic x equals lambda bold italic x rightwards double arrow left parenthesis lambda bold italic I minus bold italic A right parenthesis bold italic x equals 0 or left parenthesis bold italic A minus lambda bold italic I right parenthesis bold italic x equals 0 and for bold italic x to be a non-zero vector, det space left parenthesis lambda bold italic I minus bold italic A right parenthesis equals 0
  • The characteristic polynomial of an n cross times n matrix is:

p left parenthesis lambda right parenthesis equals det space left parenthesis lambda bold italic I minus bold italic A right parenthesis

  • In this course you will only be expected to find the characteristic equation for a 2 cross times 2 matrix and this will always be a quadratic

How do I find the characteristic polynomial?

  • STEP 1
    Write lambda bold italic I minus bold italic A, remembering that the identity matrix must be of the same order as bold italic A
  • STEP 2
    Find the determinant of lambda bold italic I minus bold italic A using the formula given to you in the formula booklet

det space bold italic A equals vertical line bold italic A vertical line equals a d minus b c

  • STEP 3
    Re-write as a polynomial 

Examiner Tip

  • You need to remember the characteristic equation as it is not given in the formula booklet

Worked example

Find the characteristic polynomial of the following matrix

bold italic A equals open parentheses table row 5 4 row 3 1 end table close parentheses.

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-1-solution

Eigenvalues & Eigenvectors

How do you find the eigenvalues of a matrix?

  • The eigenvalues of matrix bold italic A are found by solving the characteristic polynomial of the matrix
  • For this course, as the characteristic polynomial will always be a quadratic, the polynomial will always generate one of the following:
    • two real and distinct eigenvalues,
    • one real repeated eigenvalue or
    • complex eigenvalues

How do you find the eigenvectors of a matrix?

  • A value for bold italic x that satisfies the equation is an eigenvector of matrix bold italic A
  • Any scalar multiple of bold italic x will also satisfy the equation and therefore there an infinite number of eigenvectors that correspond to a particular eigenvalue
  • STEP 1
    Write bold italic x equals open parentheses table row x row y end table close parentheses
  • STEP 2
    Substitute the eigenvalues into the equation left parenthesis lambda bold italic I minus bold italic A right parenthesis bold italic x equals bold 0, and form two equations in terms of x and y
  • STEP 3
    There will be an infinite number of solutions to the equations, so choose one by letting one of the variables be equal to 1 and using that to find the other variable

Examiner Tip

  • You can do a quick check on your calculated eigenvalues as the values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix

Worked example

Find the eigenvalues and associated eigenvectors for the following matrices.

a)
bold italic A equals open parentheses table row 5 4 row 3 1 end table close parentheses  .

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-2ai-solution

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-2aii-solution

b)
bold italic B equals open parentheses table row 1 cell negative 5 end cell row 2 3 end table close parentheses .

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-2bi-solution

1-8-1-ib-ai-hl-eigenvalues--eigenvectors-we-2bii-solution

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.