Q42E Find all the eigenvalues and 鈥�... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q42E (page 358)

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.
$L (A) = A - A^{T}$ . Is $R^{2 脳 2} {toR}^{2 脳 2}$ diagonalizable?

Short Answer

Expert verified

L is diagonalizable and the eigenvalues and eigenvectors of the given linear transformation is,

$位_{1} = - 2, E_{2} = span (\{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\}) 位_{2} = 0, E_{2} = span (\{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\})$

Step by step solution

Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

$A \overset{鈫�}{x} = 位 \overset{鈫�}{x},$ here $\overset{鈫�}{x}$ is eigenvector and $位$ is the eigenvalue.

Use the formula and determine the possible values

Consider the given equation,

$L (A) = A - A^{T}$

Solve,

$L (A) = 位 A A - A^{T} = 位 A A^{T} = (位 + 1) A$

Hence, due to the entries in the main diagonal, this is only possible for $位_{1, 2} = - 2, 0$

Thus, for $位 = - 2$

$A^{T} = - A A = [\begin{matrix} a & b \\ - b & d \end{matrix}]$

$E_{2} = span (\{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\})$

Calculate for, $位 = 0$

role="math" localid="1659597245438" $A^{T} = A A = [\begin{matrix} a & b \\ b & d \end{matrix}] E_{2} = span (\{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\})$

Since, L is diagonalizable.

Therefore, the eigenvalues and eigenvectors are,

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91影视

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.
$L (A) = A - A^{T}$ . Is $R^{2 脳 2} {toR}^{2 脳 2}$ diagonalizable?

Short Answer

Step by step solution

Define eigenvalues

Use the formula and determine the possible values

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91影视

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.L(A)=A-AT. IsR2脳2toR2脳2 diagonalizable?

Short Answer

Step by step solution

Define eigenvalues

Use the formula and determine the possible values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Find all the eigenvalues and 鈥渆igenvectors鈥� of the linear transformations.
$L (A) = A - A^{T}$ . Is $R^{2 脳 2} {toR}^{2 脳 2}$ diagonalizable?