Answered: In each of the following answer true if…

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

In each of the following answer true if the statement is
always true and false otherwise. In the case of a true
statement, explain or prove your answer. In the case
of a false statement, give an example to show that the
statement is not always true. If A is a 4 × 4 matrix of rank 3 and λ = 0 is an
eigenvalue of multiplicity 3, then A is diagonalizable.

Expert Solution

Step 1

Given: $A$ is a $4 \times 4$ matrix of rank $= 3$ ; $One Eigenvalue, λ = 0$
We need to check whether the given $4 \times 4$ matrix, $A$ is diagonalizable or not.

Step 2

Let the other two eigenvalues of matrix $A {be λ}_{1} {and λ}_{2} .$

Without the loss of generality, let us assume that $λ_{1} {≠λ}_{2}$ .
Also, $λ_{1} {, λ}_{2} ≠0$ . Then there are two possible choices for Jordan canonical forms of $A$ and are given as -

$J_{1} = [\begin{matrix} λ_{1} & 0 & 0 & 0 \\ 0 & λ_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$ and $J_{2} = [\begin{matrix} λ_{1} & 0 & 0 & 0 \\ 0 & λ_{2} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}]$

Here, clearly $rank, ρ (A) = 3$ which implies that $J_{2}$ is Jordan canonical form of $A$ .
Therefore, the minimal polynomial of $A$ is : $m_{A} (λ) = (λ - λ_{1}) (λ - λ_{2}) λ^{2}$

Step by step

Solved in 3 steps

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