Find a 3 times 3 matrix A that has eigenvalues lambda=0 , 4 ,-4 with corresponding eigenvectorsbegin{bmatrix}0 1-1 end{bmatrix},begin{bmatrix}1 -11 end{bmatrix},begin{bmatrix}0 11 end{bmatrix}

Falak Kinney 2021-03-02 Answered

Find a 3×3 matrix A that has eigenvalues λ=0 , 4 ,-4 with corresponding eigenvectors
[011],[111],[011]

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Expert Answer

crocolylec
Answered 2021-03-03 Author has 100 answers

Step 1
We need to find a matrix A which has eigenvalues λ=0,4,4 and eigenvectors as
[011],[111],[011]
Now, a matrix is similar to a diagonal matrix D if there exist matrices P and P1 such that A=PDP1
Here, diagonal matrix D consists of eigenvalues as the diagonal entries.
And the matrix P is the matrix that has eigenvectors as its columns.
Hence,
D=[000040004] and 
P=[010111111]
Step 2
Hence, the matrix A is
A=PDP1
=[010111111][000040004][010111111]1
Before finding A we need to find P1=[010111111]1
Now, inverse of a matrix P is P1=adj(P)|P|
Now,
adj(P)=[+(1×11×1)(1×11×(1))+(1×1(1)×(1))(1×10×(1))+(0×10×(1))(0×11×(1))+(1×10×(1))(0×10×1)+(0×(1)1×1)]T
=[220101101]T
=[211200011]
And
|P|=|010111111|
=0|1111||1111|+0|1111|=2
Hence,

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Jeffrey Jordon
Answered 2022-01-30 Author has 2087 answers

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