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(i) Let a, b and c be the last three non-zero digits of your enrolment number. Use them to form the a ba matrix M = b b с b a with a brief justification of why these must be true. State two facts you can immediately deduce about the eigenvalues of M, a

(i) Let a, b and c be the last three non-zero digits of your enrolment number. Use them to form the a ba matrix M = b b с b a with a brief justification of why these must be true. State two facts you can immediately deduce about the eigenvalues of M, a

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 27EQ
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A=1 B=5 C=1

(i) Let a, b and c be the last three non-zero digits of your enrolment number. Use them to form the a b a b c b State two facts you can immediately deduce about the eigenvalues of M, b a a matrix M = with a brief justification of why these must be true. (ii) Find the characteristic equation of M and show that M satisfies the Cayley-Hamilton Theorem. (iii) Calculate the eigenvalues of M, and a corresponding eigenvector for each eigenvalue. (iv) Hence write down a diagonalised form of M. (v) State Gershgorin's Circle Theorem and show that the eigenvalues of M satisfy the theorem. (vi) Why does Gershgorin's Circle Theorem imply that the eigenvalues of a diagonal matrix with distinct non-zero entries on the main diagonal are simply the diagonal elements? (11) and (19) (vii) Show that the matrices trace, determinant and eigenvalues. cannot be similar but that they do have the same (viii) Give the possible eigenvalue(s) of an n x n projection matrix, that projects onto a subspace of dimension m. What are the algebraic multiplicity and geometric multiplicity of each eigenvalue? Hint: A projection matrix P satisfies the relation P² = P.
Transcribed Image Text:(i) Let a, b and c be the last three non-zero digits of your enrolment number. Use them to form the a b a b c b State two facts you can immediately deduce about the eigenvalues of M, b a a matrix M = with a brief justification of why these must be true. (ii) Find the characteristic equation of M and show that M satisfies the Cayley-Hamilton Theorem. (iii) Calculate the eigenvalues of M, and a corresponding eigenvector for each eigenvalue. (iv) Hence write down a diagonalised form of M. (v) State Gershgorin's Circle Theorem and show that the eigenvalues of M satisfy the theorem. (vi) Why does Gershgorin's Circle Theorem imply that the eigenvalues of a diagonal matrix with distinct non-zero entries on the main diagonal are simply the diagonal elements? (11) and (19) (vii) Show that the matrices trace, determinant and eigenvalues. cannot be similar but that they do have the same (viii) Give the possible eigenvalue(s) of an n x n projection matrix, that projects onto a subspace of dimension m. What are the algebraic multiplicity and geometric multiplicity of each eigenvalue? Hint: A projection matrix P satisfies the relation P² = P.
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can you write this all on paper if possible including all parts that have been answered

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can you continue the question pls and continue doing it on paper many thanks.

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