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kini.Amith
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given that 2 matrices have the same eigenvalues is it necessary that they be similar? If not, what is the connection between those 2?
You have asked that repeatedly. Please tell us what you mean by "in common"!kini.Amith said:i nderstand that they need not be similar, but then what do they have in common?
HallsofIvy said:No. Two matrices are similar if and only if they have the same eigenvalues and corresponding eigenvectors.
stringy said:Would you mind clarifying this point? It's well known that a similarity transformation preserves the spectrum, but the eigenvectors?
The matrices
[ 0 1 ]
[ 0 0 ]
and
[ 0 0 ]
[ 1 0 ]
are similar via the permutation matrix
[ 0 1 ]
[ 1 0 ],
but they don't share the same eigenvectors.
Eigenvalues are a mathematical concept used in linear algebra to describe the properties of a matrix. They are defined as the scalar values that, when multiplied by a vector, result in the same vector. In other words, they represent the scaling factor of a vector when it is transformed by a matrix.
To determine if two matrices have the same eigenvalues, you can calculate the determinant of both matrices. If the determinant is the same for both matrices, then they have the same eigenvalues. Additionally, you can also compare the characteristic polynomial of the matrices, as they will have the same roots if the matrices have the same eigenvalues.
Having the same eigenvalues means that the matrices have similar properties and can be transformed in a similar way. This is useful in various applications, such as in solving systems of linear equations, finding the principal components of a dataset, and in understanding the behavior of dynamical systems.
Yes, it is possible for two matrices to have the same eigenvalues but different eigenvectors. This means that although the matrices have the same scaling properties, they may transform vectors in different directions. This is because the eigenvectors are not unique and can be scaled or rotated.
The diagonalization of a matrix involves finding a diagonal matrix that is similar to the original matrix. In this process, the eigenvalues of the original matrix are the diagonal elements of the diagonal matrix. Therefore, two matrices with the same eigenvalues are easier to diagonalize and can be transformed into the same diagonal matrix.