Week 9: Part 1: Show that there is no relationship between any kind of row operation and the eigenvalues of the matrices involved as follows. For each of the three types of row operation cR; + R; → R;, cR; → Rj, and R; + R; which are adding two rows, multiplying a row by a scalar, and switching two rows: Find four matrices (for a total of 12 matrices) A, B, C, D such that A B and C # D and also A ~ B and C ~ D, the matrix A is row equivalent to B via exactly one row operation of the appropriate type, and also C is row equivalent to D via exactly one row operation of the appropriate type, and such that A and B have the exact same eigenvalues but C and D have different eigenvalues, and be sure to state all eigenvalues. Conclude that there is no relationship whatsoever between two matrices "being row equivalent" and "have the same eigenvalues".
Week 9: Part 1: Show that there is no relationship between any kind of row operation and the eigenvalues of the matrices involved as follows. For each of the three types of row operation cR; + R; → R;, cR; → Rj, and R; + R; which are adding two rows, multiplying a row by a scalar, and switching two rows: Find four matrices (for a total of 12 matrices) A, B, C, D such that A B and C # D and also A ~ B and C ~ D, the matrix A is row equivalent to B via exactly one row operation of the appropriate type, and also C is row equivalent to D via exactly one row operation of the appropriate type, and such that A and B have the exact same eigenvalues but C and D have different eigenvalues, and be sure to state all eigenvalues. Conclude that there is no relationship whatsoever between two matrices "being row equivalent" and "have the same eigenvalues".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Week 9: Part 1: Show that there is no relationship between any kind of row operation and the
eigenvalues of the matrices involved as follows. For each of the three types of row operation
cR; + R; → Rj, cR; → Rj, and R; + R; which are adding two rows, multiplying a row by a
scalar, and switching two rows: Find four matrices (for a total of 12 matrices) A, B,C, D
such that A 7 B and C D and also A ~
to B via exactly one row operation of the appropriate type, and also C is row equivalent to
B and C ~
D, the matrix A is row equivalent
D via exactly one row operation of the appropriate type, and such that A and B have the
exact same eigenvalues but C and D have different eigenvalues, and be sure to state all
eigenvalues. Conclude that there is no relationship whatsoever between two matrices “being
row equivalent" and "have the same eigenvalues".
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