Question

Which of the following are NECESSARY CONDITIONS for an n x n matrix A to be diagonalizable?

i) A has n distinct eigenvalues

ii) A has n linearly independent eigenvectors

iii) The algebraic multiplicity of each eigenvalue equals its geometric multiplicity

iv) A is invertible

v) The columns of A are linearly independent

NOTE: The answer is more than 1 option.

Answer #1

Let A be a matrix with an eigenvalue λ that has an algebraic
multiplicity of k, but a geometric multiplicity of p < k, i.e.
there are p linearly independent generalised eigenvectors of rank 1
associated with the eigenvalue λ, equivalently, the eigenspace of λ
has a dimension of p. Show that the generalised eigenspace of rank
2 has at most dimension 2p.

The questions this week is about diagonalizability of matrices
when we have fewer than n different eigenvalues. Recall the
following facts as starting points:
• An n × n matrix is diagonalizable if and only if it has n
linearly independent eigenvectors.
• Eigenvectors with different eigenvalues must be linearly
independent.
• The number of times an eigenvalue appears as a root of the
characteristic polynomial is at least the dimension of the
corresponding eigenspace, and the total degree of...

5. Suppose A is an n × n matrix, whose entries are all real
numbers, that has n distinct real eigenvalues. Explain why R n has
a basis consisting of eigenvectors of A. Hint: use the “eigenspaces
are independent” lemma stated in class.
6. Unlike the previous problem, let A be a 2 × 2 matrix, whose
entries are all real numbers, with only 1 eigenvalue λ. (Note: λ
must be real, but don’t worry about why this is true)....

True or False: An n X n matrix A has n linearly
independent eigenvectors. Justify your answer
(explain)
These questions follow the exercises in Chapter 4 of the book,
Scientific Computing: An Introduction, by Michael Heath. They study
some of the properties of eigenvalues, eigenvectors, and some ways
to compute them.

True or False:
An n X n matrix A has n linearly independent eigenvectors.
Justify your answer (explain)
These questions follow
the exercises in Chapter 4 of the book, Scientic Computing: An
Introduction, by Michael Heath. They study some of the properties
of eigenvalues, eigenvectors, and some ways to compute them.

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

2
1
1
1
0
1
1
1
0
These questions have got me confused:
1. By calculation, I know this matrix has eigenvalue -1, 0, 3
and they are distinct eigenvalues. Can I directly say that this
matrix is diagonalizable without calculating the eigenspace and
eigenvectors? For all situations, If we get n number of answers
from (aλ+b)n , can we directly ensure that the matrix is
diagonalizable?
2. My professor uses CA(x)=det(λI-A) but the textbook
shows CA(x)=det(λI-A). which...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Which one of the following are zero-coupon bonds?
I) Treasury bill
II) Treasury note
III) Treasury bond
IV) Commercial paper
V) Agency bonds
I, V
I, II, III
I, IV, V
II, III
I, IV
You buy a call option on Citibank with the strike price of 100.
Suppose the Citibank's stock price is 110 on the option expiration
date. What is your payoff?
0.
10.
20.
-10.
-20.
According the lectures, what one of the following signals can be...

Let a, b, and n be integers with n > 1 and (a, n) = d.
Then
(i)First prove that the equation a·x=b has solutions in n if and
only if d|b.
(ii) Next, prove that each of u, u+n′, u+ 2n′, . . . , u+
(d−1)n′ is a solution. Here,u is any particular solution guaranteed
by (i), and n′=n/d.
(iii) Show that the solutions listed above are distinct.
(iv) Let v be any solution. Prove that v=u+kn′ for...

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