If A is any symmetric 3x3 matrix with eigenvalues -2,3 , and 4 , and is a unit vector in , what are the possible values of the dot product ?
The possible values of the dot product are , where is a unit vector
In linear algebra, a symmetric matrix is a square matrix that stays unchanged when its transpose is calculated. In that instance, a symmetric matrix is one whose transpose equals the matrix itself.
Given that, A is any symmetric 3x3 matrix with eigenvalues -2,3 and 4 , and is a unit vector in . From the spectral theorem, we know that there exists an orthonormal eigenbasis , with associated real eigenvalues and (Arrange things so that ). Now consider unit vector as represented below:
Now evaluate as follows:
From (1),(2) and (3) we can imply that the possible values of the dot product areas represented below:
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