03/07/2020 Basic Linear Algebra - - Unit 9 - Week 7
https://onlinecourses.nptel.ac.in/noc20_ma08/unit?unit=49&assessment=96 1/2
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» Basic Linear Algebra (course)Unit 9 - Week 7
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Course outline
How does an NPTEL online course work?
Week 0 - Welcome to the course!
Week 1 Week 2 Week 3 Week 4 Week 5 Week 6
Week 7 Lecture 34 : Diagonalization and its Applications I (unit?
unit=49&lesson=57)
Due on 2020-04-15, 23:59 IST.
1 point 1)
1 point 2)
Assignment 7 - Objective
The due date for submitting this assignment has passed.
As per our records you have not submitted this assignment.
State whether True or False.
Let be an real matrix diagonalizable over and be an real matrix such that , for some invertible real matrix . Then is diagonalizable.
True False
No, the answer is incorrect.
Score: 0
Accepted Answers:
True
Let be an real matrix which is diagonalizable over . Let
be a polynomial function with such that , whenever in . Let be the distinct eigenvalues of . Then
The matrix is diagonalizable over with distinct eigenvalues , and the algebraic multiplicity of each is equal to the geometric multiplicity of .
The matrix is diagonalizable over if and only if .
The matrix is diagonalizable over , but the distinct eigenvalues of need not be .
The matrix need not be diagonalizable over .
N h i i
𝐴 𝑛 × 𝑛 ℝ 𝐵 𝑛 × 𝑛
𝐵 = 𝑈
−1𝐴𝑈 𝑛 × 𝑛 𝑈 𝐵
𝐴 𝑛 × 𝑛 ℝ
𝑃(𝑡) = 𝑎
0+ 𝑡 + ⋯ + 𝑎
1𝑎
𝑘𝑡
𝑘𝑎
0, … , 𝑎
𝑘∈ ℝ 𝑃(𝑠) < 𝑃(𝑡) 𝑠 < 𝑡 ℝ 𝜆
1, … , 𝜆
𝑚𝐴
𝑃(𝐴) ℝ 𝑃( ), … , 𝑃( ) 𝜆
1𝜆
𝑚𝜆
𝑖𝑃( ) 𝜆
𝑖𝑃(𝐴) ℝ deg 𝑃 ≤ 1
𝑃(𝐴) ℝ 𝑃(𝐴)
𝑃( ), … , 𝑃( ) 𝜆
1𝜆
𝑚𝑃(𝐴) ℝ
03/07/2020 Basic Linear Algebra - - Unit 9 - Week 7
https://onlinecourses.nptel.ac.in/noc20_ma08/unit?unit=49&assessment=96 2/2
Week 8 Lecture 35 : Diagonalization and its Applications II (unit?
unit=49&lesson=58) Lecture 36 : Diagonalization and its Applications III (unit?
unit=49&lesson=59) Weekly
Feedback (unit?
unit=49&lesson=80) Download
Videos (unit?
unit=49&lesson=89) Quiz :
Assignment 7 - Objective (assessment?
name=96)
1 point 3)
1 point 4)
No, the answer is incorrect.
Score: 0
Accepted Answers:
The matrix is diagonalizable over with distinct eigenvalues , and the algebraic multiplicity of each is equal to the geometric multiplicity of .
State whether True or False.
Let be two real matrices which are diagonalizable over such that they have the same set of distinct eigenvalues with the same set of corresponding algebraic multiplicities, that is, each eigenvalue has the same algebraic multiplicity as an eigenvalue of and as an eigenvalue of . Then there exists an invertible real matrix such that .
True False
No, the answer is incorrect.
Score: 0
Accepted Answers:
True
State whether True of False.
There is no nonzero real matrix which is diagonalizable over such that . True
False
No, the answer is incorrect.
Score: 0
Accepted Answers:
False
