Prompt 4: Power Method for Eigenvalues and Eigenvectors The power method is an iterative method used...

Transcribed Image Text:

Prompt 4: Power Method for Eigenvalues and Eigenvectors The power method is an iterative method used to extract the dominant eigenvalue (largest eigen- value in absolute value) and corresponding eigenvector from a matrix A. The method relies on successive multiplication of a non-zero vector by A using the following iterative scheme: Ark max (AFK) where To, the initial vector, can be pretty much any non-zero vector. The denominator max (AF) is the maximum absolute value entry of the current iteration, and dividing by this is meant to keep the magnitude of the eigenvector small through successive iterations. If the matrix has a dominant eigenvalue A with an associated eigenvector , then the sequence converges to 7. Let A be given by: A = Tk+1 = -242 278 209 -110 -84 444 -492 -352 232 132 -288 304 248 -112 -96 642 -742 -511 362 192 702 -802 -559 370 220 a.) Using the above method, estimate the dominant eigenvector 50 of matrix A. b.) Using the relationship AF = X for an eigenvalue/eigenvector pair (A,) to find the asso- ciated eigenvalue for your eigenvector found in (a.). c.) The eigenvalues of the inverse of a matrix A¹ are the multiplicative inverses of the eigen- values of A and are paired with the same eigenvectors. This is illustrated using the following calculation for an eigenvalue/eigenvector pair (A, 2): Let Az = X ⇒ A¹A = A-¹X ⇒ IZ = XA-¹ ⇒ = XA¹7 7 = A-¹7 ⇒ (1/A, 2) is an eigenvalue/eigenvector pair of A-¹ ← Construct A-¹ (you do not need to supply the inverse in the write up but describe how you got it) and use the power method to find the smallest magnitude eigenvalue and its associated eigenvector y of matrix A. Prompt 4: Power Method for Eigenvalues and Eigenvectors The power method is an iterative method used to extract the dominant eigenvalue (largest eigen- value in absolute value) and corresponding eigenvector from a matrix A. The method relies on successive multiplication of a non-zero vector by A using the following iterative scheme: Ark max (AFK) where To, the initial vector, can be pretty much any non-zero vector. The denominator max (AF) is the maximum absolute value entry of the current iteration, and dividing by this is meant to keep the magnitude of the eigenvector small through successive iterations. If the matrix has a dominant eigenvalue A with an associated eigenvector , then the sequence converges to 7. Let A be given by: A = Tk+1 = -242 278 209 -110 -84 444 -492 -352 232 132 -288 304 248 -112 -96 642 -742 -511 362 192 702 -802 -559 370 220 a.) Using the above method, estimate the dominant eigenvector 50 of matrix A. b.) Using the relationship AF = X for an eigenvalue/eigenvector pair (A,) to find the asso- ciated eigenvalue for your eigenvector found in (a.). c.) The eigenvalues of the inverse of a matrix A¹ are the multiplicative inverses of the eigen- values of A and are paired with the same eigenvectors. This is illustrated using the following calculation for an eigenvalue/eigenvector pair (A, 2): Let Az = X ⇒ A¹A = A-¹X ⇒ IZ = XA-¹ ⇒ = XA¹7 7 = A-¹7 ⇒ (1/A, 2) is an eigenvalue/eigenvector pair of A-¹ ← Construct A-¹ (you do not need to supply the inverse in the write up but describe how you got it) and use the power method to find the smallest magnitude eigenvalue and its associated eigenvector y of matrix A.

Answer rating: 100% (QA)

a Using the above method estimate the dominant eigenvector 50 of matrix A The power method is used to estimate the dominant eigenvector x of matrix A The initial vector To can be any nonzero vector We ... View the full answer