The top matrix on the menu is the diagonal matrix
Initially, when you select this matrix, the
To find:
By rotate x counterclockwise until x and Ax are parallel, the eigenvalues and eigenvectors of a
Answer to Problem 1E
In first figure, the eigenvalue is
In second figure, the eigenvalue is
In third figure the eigenvalue is
In third figure, the eigenvalue is
The eigenvalues are
The eigenvectors are
Explanation of Solution
Given:
The diagonal matrix
the vectors x and Ax should both be aligned along the positive x-axis.
Calculation:
Now using the eigshow command in MATLAB, we draw the vectors “x” and “Ax” as follows:
From the figure, we see that the vectors x and Ax both are aligned along the positive x-axis.
As both the vectors are parallel, one vector is multiple of the other.
Therefore,
From the figure above we obtain as:
Therefore,
Thus for this position the eigenvalue is
Now, rotate x counterclockwise until x and Ax are parallel as follows:
Again, as both the vectors are parallel, one vector is multiple of the other.
Hence,
From the second figure above it is obtain as:
Therefore,
Thus for this position the eigenvalue is
Let the matrix:
Now using the eigshow command in MATLAB, we draw the vectors x and Ax as follows:
Since both the vectors are parallel, we can say that one vector is multiple of the other.
So,
From the figure above we obtain as:
Therefore,
Thus for this position the eigenvalue is
Now, we rotate x counterclockwise until x and Ax are parallel as follows:
Again, as both the vectors are parallel, one vector is multiple of the other.
Hence,
From the second figure above it is obtain as:
Hence,
Thus for this position the eigenvalue is
The eigenvalues are
This experiment also works for a
Conclusion:
In first figure, the eigenvalue is
In second figure, the eigenvalue is
In third figure the eigenvalue is
In third figure, the eigenvalue is
The eigenvalues are
the eigenvectors are
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