Q28E Consider the matrixJn(k)=[k10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: Q28E (page 345)

Consider the matrix
$J_{n} (k) = [\begin{matrix} k & 1 & 0 & 鈥� & 0 & 0 \\ 0 & k & 1 & 鈥� & 0 & 0 \\ 0 & 0 & k & 鈥� & 0 & 0 \\ M & M & M & 0 & M & M \\ 0 & 0 & 0 & 鈥� & k & 1 \\ 0 & 0 & 0 & 鈥� & 0 & k \end{matrix}]$
(with all k鈥檚 on the diagonal and 1鈥檚 directly above), where k is an arbitrary constant. Find the eigenvalue(s) of Jn(k), and determine their algebraic and geometric multiplicities.

Short Answer

Expert verified

$位_{1, . . ., n} = k, almu (k) = n, gemu (k) = 1$

Step by step solution

Finding eigen values and Definition of almu and gemu

"Algebraic multiplicity" and "geometric multiplicity" are abbreviated as "almu" and "gemu," respectively.

Since this matrix is upper triangular, its eigenvalues are the entries on its main diagonal, which are $位_{1, . . ., n} = k$ .From this, we directly conclude that almu(k) = n

Solving (A - kI) = 0 using λ=k and finding gemu(k)

For $位 = k$ ,we solve

$(A - Kl) = 0$

localid="1659595213510" $[\begin{matrix} 0 & 1 & 0 & 鈥� & 0 & 0 \\ 0 & 0 & 1 & 鈥� & 0 & 0 \\ 0 & 0 & 0 & 鈥� & 0 & 0 \\ 鈰� & 鈰� & 鈰� & 鈰� & 鈰� & 鈰� \\ 0 & 0 & 0 & 鈥� & 0 & 1 \\ 0 & 0 & 0 & 鈥� & 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ 鈰� \\ x_{n - 1} \\ x_{n} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \\ 鈰� \\ 0 \\ 0 \end{matrix}] x_{2} = x_{3} = 鈥� x_{n - 1} = x_{n} = 0$

$E_{k} = span ((\begin{matrix} 1 \\ 0 \\ 0 \\ 鈰� \\ 0 \\ 0 \end{matrix}))$

This means gemu(k)=1

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Short Answer

Step by step solution

Finding eigen values and Definition of almu and gemu

Solving (A - kI) = 0 using λ=k and finding gemu(k)

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91影视

Consider the matrixJn(k)=[k10鈥�000k1鈥�0000k鈥�00MMM0MM000鈥�k1000鈥�0k](with all k鈥檚 on the diagonal and 1鈥檚 directly above), where k is an arbitrary constant. Find the eigenvalue(s) of Jn(k), and determine their algebraic and geometric multiplicities.

Short Answer

Step by step solution

Finding eigen values and Definition of almu and gemu

Solving (A - kI) = 0 using λ=k and finding gemu(k)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Discrete Mathematics

Mechanics Maths

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.