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IBMathsDPApplications & Interpretation (AI)HLFlashcards1. Number & AlgebraEigenvalues & Eigenvectors

Eigenvalues & Eigenvectors (DP IB Applications & Interpretation (AI))

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  • What is an eigenvalue?

    For a square 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. 

  • What is an eigenvector?

    For a square 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 bold italic x is an eigenvector corresponding to the eigenvalue lambda.

  • What is the characteristic polynomial of a matrix?

    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

    Where:

    • bold italic A is a square matrix

    • lambda is an eigenvalue of the matrix bold italic A

    • bold italic x is eigenvector corresponding to the eigenvalue lambda

    This equation is not given in your exam formula booklet.

  • What is the first step in finding the characteristic polynomial of a matrix, e.g. bold italic A equals open parentheses table row 2 7 row cell negative 3 end cell 4 end table close parentheses?

    The first step in finding the characteristic polynomial of a matrix is to write down lambda bold italic I minus bold italic A.

    E.g. for the matrix bold italic A equals open parentheses table row 2 7 row cell negative 3 end cell 4 end table close parentheses, write down lambda open parentheses table row 1 0 row 0 1 end table close parentheses minus open parentheses table row 2 7 row cell negative 3 end cell 4 end table close parentheses.

    Remember that bold italic I must be of the same order as bold italic A.

  • What is the next step in finding the characteristic polynomial of a matrix, after writing down lambda bold italic I minus bold italic A?

    The next step in finding the characteristic polynomial of a matrix, after writing down lambda bold italic I minus bold italic A, is to find its determinant, det open parentheses lambda bold italic I minus bold italic A close parentheses.

  • True or False?

    The characteristic polynomial of a square matrix will always be a quadratic.

    True.

    The characteristic polynomial of a square matrix will always be a quadratic in this course.

  • When solving the characteristic polynomial of a matrix, what possible solutions could there be?

    When solving the characteristic polynomial of a matrix, there could be:

    • two real eigenvalues,

    • one real, repeated eigenvalue,

    • or two complex eigenvalues.

  • How do you find the eigenvectors of a matrix?

    To find the eigenvectors of a matrix:

    1. Write bold italic x equals open parentheses table row x row y end table close parentheses.

    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.

    3. Let one of the variables be equal to 1 and use that to find the other variable.

  • True or False?

    There are a finite number of eigenvectors that correspond to a particular eigenvalue.

    False.

    There are an infinite number of eigenvectors that correspond to a particular eigenvalue.

    This is because for any specific value of bold italic x that satisfies the equation left parenthesis lambda bold italic I minus bold italic A right parenthesis bold italic x equals bold 0, any scalar multiple of bold italic x will also satisfy the equation.

  • True or False?

    The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix.

    True.

    The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix.

    This can be used as a quick check of your calculated eigenvalues.

  • What is a diagonal matrix?

    A non-zero, square matrix is considered to be diagonal if all elements not along its leading diagonal are zero.

  • How can bold italic P be used to diagonalise a matrix bold italic M, given that bold italic P equals left parenthesis bold italic x subscript 1 bold italic x subscript 2 right parenthesis, where the first column is the eigenvector bold italic x subscript 1and the second column is the eigenvector bold italic x subscript 2?

    If the matrix bold italic M is pre-multiplied by bold italic P and post-multiplied by bold italic P to the power of negative 1 end exponent, then it will result in the diagonal matrix of eigenvalues, bold italic D equals open parentheses table row cell lambda subscript 1 end cell 0 row 0 cell lambda subscript 2 end cell end table close parentheses.

    bold italic D equals bold italic P to the power of negative 1 end exponent bold italic M bold italic P

  • What is one of the main applications of diagonalising a matrix?

    One of the main applications of diagonalising a matrix is to make it easy to find powers of the matrix.

  • How do you find higher powers of a diagonalised square matrix, e.g. find bold italic M to the power of n, given bold italic M equals open parentheses table row a 0 row 0 b end table close parentheses.

    To find higher powers of a diagonalised square matrix, raise each element along the leading diagonal of the matrix to the given power.

    E.g. if bold italic M equals open parentheses table row a 0 row 0 b end table close parentheses, then bold italic M to the power of bold n equals open parentheses table row a 0 row 0 b end table close parentheses to the power of n equals open parentheses table row cell a to the power of n end cell 0 row 0 cell b to the power of n end cell end table close parentheses.

  • What is the power formula for a matrix?

    The power formula for a matrix is bold italic M to the power of n equals bold italic P bold italic D to the power of n bold italic P to the power of negative 1 end exponent

    Where:

    • bold italic M is a matrix

    • bold italic P is the matrix of eigenvectors

    • bold italic D is the matrix of eigenvalues

    This formula is given in your exam formula booklet.

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