Answered: 0 4 -4 A = -4 0 -2 4 2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

deal with the eigenvalue/eigenvector problem for n×n real skew-symmetric matrices.

Q.Determine all eigenvalues and corresponding eigenvectors of the matrix

Expert Solution

Step 1: Objective:

To determine all eigenvalues and corresponding eigenvectors of the matrix.

$open square brackets table row 0 4 cell negative 4 end cell row cell negative 4 end cell 0 cell negative 2 end cell row 4 2 0 end table close square brackets$

Step 2: Finding eigenvalues.

Let k denote the eigenvalues.

Then.

$table attributes columnalign right center left columnspacing 0px end attributes row cell vertical line A minus k I vertical line end cell equals cell open vertical bar table row cell negative k end cell 4 cell negative 4 end cell row cell negative 4 end cell cell negative k end cell cell negative 2 end cell row 4 2 cell negative k end cell end table close vertical bar end cell row blank equals cell equals open parentheses negative k close parentheses open vertical bar table row cell negative k end cell cell negative 2 end cell row 2 cell negative k end cell end table close vertical bar minus 4 times open vertical bar table row cell negative 4 end cell cell negative 2 end cell row 4 cell negative k end cell end table close vertical bar plus open parentheses negative 4 close parentheses open vertical bar table row cell negative 4 end cell cell negative k end cell row 4 2 end table close vertical bar end cell row blank equals cell open parentheses negative k close parentheses open parentheses k squared plus 4 close parentheses minus 4 open parentheses 4 k plus 8 close parentheses plus open parentheses negative 4 close parentheses open parentheses negative 8 plus 4 k close parentheses end cell row blank equals cell negative k cubed minus 36 k end cell end table$

Consider $vertical line A minus k I vertical line equals 0$

Thus,

$table attributes columnalign right center left columnspacing 0px end attributes row cell negative k cubed minus 36 k end cell equals 0 row cell negative k open parentheses k squared plus 36 close parentheses end cell equals 0 row k equals cell 0 space o r space k equals plus-or-minus 6 i end cell end table$

Eigenvalues: $0 comma space 6 i comma space minus 6 i$

Step by step

Solved in 4 steps with 40 images

Knowledge Booster

$Matrix Eigenvalues and Eigenvectors$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions