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Category  Math 

Subject  Algebra 
Difficulty  Undergraduate 
Status  Solved 
More Info  Calculus Homework Help 
Short Assignment Requirements
Hi, there is some problem with the matrix. Need to find eigenvalues, eigenvectors, chains of eigenvectors and create diagonalization matrix G and compute G^1 *D*G. See details in the attachment. Solve them step by step, please!
Assignment Image
For the following matrices, find:
● All eigenvalues and state their multiplicity.
• As many independent eigenvectors as possible.
• A complete set of generalized eigenvectors.
• Describe all of the chains of eigenvectors.
A
2 0 0
0 2 0
0
2
0 0 2
and
0
2
1 0 0
2 0
0
0

0
2 1
0 2
A=
B =
D =
Finally, use the (generalized) eigenvalues that you found for D to create the ‘diagonal
ization' matrix G and compute G¹DG.
A note on inverses: Given an invertible matrix
A21 = a12a33  a13a32,
A31 = a12a23  a13a22,
A¹
a11 a12 a13
a23
a21 a22
a31 a32 a33,
we can use the adjugate matrix Adj(A) to define the inverse of A, where
A11
 A21
A31
Adj (A) =
A12
A22
 A32
A13  A23
A33
and A¡¡ is the determinant of the 2 × 2 matrix we get from deleting the ith row and jth
column in A. For example,
0
1
det (A)
0 0
1
2
0
0
2
0 0 2
2 −1
0
2
0 0 2
 Adj(A).
2
A22 = a11a33  a13a31.
Note that this is slightly different from what I wrote in lecture. This is correct, while
the lecture was wrong. (Precisely: this is the transpose of what I wrote in lecture.)
With this definition,