Question

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 C_{A}(x)=det(λI-A) but the textbook
shows C_{A}(x)=det(λI-A). which one is correct?

Thank you very much.

Answer #1

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...

Which of the following are NECESSARY CONDITIONS for an n x n
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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.

Matrix A is given as A =
0 2 −1
−1 3 −1
−2 4 −1
a) Find all eigenvalues of A.
b) Find a basis for each eigenspace of A.
c) Determine whether A is diagonalizable. If it is, ﬁnd an
invertible matrix P and a diagonal matrix D such that D =
P^−1AP.
Please show all work and steps clearly please so I can follow
your logic and learn to solve similar ones myself. I...

Suppose that a 4 × 4 matrix A has eigenvalues
?1 = 1, ?2 = ? 2,
?3 = 4, and ?4 = ? 4. Use
the following method to find det (A).
If
p(?) = det (?I ? A) =
?n +
c1?n ? 1 + ? +
cn
So, on setting ? = 0, we obtain that
det (? A) = cn or det (A)
= (? 1)ncn
det (A) =

What are the eigenvectors and eigenvalues of the matrix A = [ 0 1
1 1]
how can i write the vector v = [1 0] as a linear combition of
the eigenvectors?
please explain

6. Let A = 3 −12 4 −1 0 −2 −1 5 −1 . 1 (a) Find all
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eigenvalue of A n for any integer n.). (b) Determine whether A is
invertible (Check if the constant term of the characteristic
polynomial χA(λ) is non-zero.). (c) If A is invertible, find (i)
A−1 using the Cayley-Hamilton theorem (ii) All the eigenvalues...

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
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c) The solution set of b is the set of all vectors of the form u =
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where vh is any solution...

the Markov chain on S = {1, 2, 3} with transition matrix
p is
0 1 0
0 1/2 1/2;
1/2 0 1/2
We will compute the limiting behavior of pn(1,1) “by hand”.
(A) Find the three eigenvalues λ1, λ2, λ3 of p. Note: some are
complex.
(B) Deduce abstractly using linear algebra and part (A) that we
can write
pn(1, 1) = aλn1 + bλn2 + cλn3
for some constants a, b, c. Don’t find these constants yet.
(C)...

1. Given β = XT 1×nAn×nXn×1, show that the gradient of β with
respect to X has the following form: ∇β = X T (A + A T ). Also,
simplify the above result when A is symmetric. (Hint: β can be
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Let f have a power series representation, S. Suppose that
f(0)=1, f’(0)=3, f’’(0)=2 and f’’’(0)=5.
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b. If, in addition to the above information, we know that S
converges on the interval [-2,2] and that |f’’’’(x)|< 11 on that
interval, then to what degree of accuracy can we estimate
f(1)?

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