When is a rank one matrix diagonalizable? Justify your answer. What is one choice of the diagonalizing similarity? What happens when it is not diagonalizable? Justify your answer.

sjeikdom0 2021-03-09 Answered
When is a rank one matrix diagonalizable? Justify your answer. What is one choice of the diagonalizing similarity? What happens when it is not diagonalizable? Justify your answer.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Tasneem Almond
Answered 2021-03-10 Author has 91 answers

Step 1
Let, the rank of (A3I) is 1. A matrix is diagonalizable if for each eigen value λ, the rank r of the matrix
(A3I)=n-the multiplicity of λ , where n= size of the matrix
Step 2
A n×n matrix A is diagonalizable if it is similar to a diagonal matrix, that is, if there exists an invertible n×n matrix c and a diagonal matrix D such that A=CDC1
Example:
A=[100050006]=I3[100050006]I31
Ay diagonal matrix A is diagonalizable as it is similar to itself.
Step 3
If the matrix is not diagonalizable, the one has to find a matrix with same properties consisting of eigen values on the leading diagonal and either ones or zeros on the super diagonal.

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-05-17

Find the point on the line y=2x+3 that is closest to the origin.

asked 2022-07-02
We are given a ratio:
g ( x ) f ( x )
where:
g ( x ) R +
f ( x ) N f ( x ) 2
So g ( x ) returns values in [ 0 , + ] while f ( x ) returns values in { 2 , 3 , 4 , }.

I am looking for a confirmation about a very simple question: if I maximize g ( x ) f ( x ) , do I also maximize g ( x ) f ( x ) 1 in this very particular case?
asked 2022-02-08

Find the surface area of the prism. 

asked 2021-08-07
Sue is starring a lawn-cutting business. Her start-up cost tot buy two lawn mowers and an edge trimmer is $665. She has figured out that she will use about $1 in gas for each lawn. If she charges $20 per lawn what will her break-even point be?
asked 2021-08-09
Given: Concentric circles with center O; AB= 18: BE=4; COperpAB
Find: EO
asked 2021-02-23
Which shape of the net of a cylinder represents the lateral surface area of the cylinder? [Pic]
asked 2022-07-15
Finding surface area and volume of a sphere using only Pappus' Centroid Theorem
I wonder if it is possible to derive surface area and volume of a sphere seperately using techniques involving pappus' theorem.
I did some calculation and found out the ratio of surface area and volume. Here is my work,
1. My key observation is finding out that the centroid of a semidisk of raidus r is also the centroid of a semicircle of raidus 2 r / 3 when the centers coincide. (By "centers" I mean the centers of full circle and disc induced by half of them.)
2. I sliced the semidisk identical pieces of triangles(infinitely many) and by locating each triangle's centroid I form a semicircle of radius 2 r / 3 which must have the same centroid with semidisk.
3. Then let's say our centroid is located h distance above the center. We still don't know it.
4. Using the theorem, the circular path taken by the centroid of the semidisk times the area of the centroid should give the volume of the sphere of radius r. And similarly, the circular path taken by the same centroid of the semicircle times the arc length of semicircle should give the surface area of a sphere of radius 2r/3.
5. We know that when the radius increases with a proportion, corresponding surface area will also increase with the square of that proportion. Thus, we need to multiply the surface area of sphere of radius 2r/3 by the factor 9/4 to get the surface area of the sphere of radius r.
Here is the calculations,
V = 2 π h π r 2 / 2 = π 2 r 2 h
S = 9 / 4 2 π h π ( 2 r / 3 ) = 3 π 2 r h
Since we don't know h, I simply divide them to cancel it out and get, V / S = r / 3.

New questions