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Bachelier
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let A and B be n x n matrices over a field F. Suppose that A^2 = A and B^2 = B. Prove that A and B are similar if and only if they have the same rank.
arkajad said:Perhaps this will help: Every vector x can be written as x=(I-A)x+Ax. If [tex]A^2=A[/tex] then the range of (I-A) and the range of A are disjoint complementary subspaces (needs a proof). A vanishes on the first subspace while the second one consists of vectors of eigenvalues 1. Similarly for B.
If you know the eigenvalues, what can you tell about the null space and therefore the nullity? Using the nullity, what can you tell about rank given that they have the same dimensions?Bachelier said:Can I use the same eigenvalues argument?
They have +1,-1 as eigenvalue, but how do I make the connection to the rank?
Thank you
Anonymous217 said:If you know the eigenvalues, what can you tell about the null space and therefore the nullity? Using the nullity, what can you tell about rank given that they have the same dimensions?
The rank of a matrix is the maximum number of linearly independent rows or columns. To prove the rank of matrices A and B, you can use the Gaussian elimination method to reduce the matrices into their row-echelon form. The number of non-zero rows in the row-echelon form will be the rank of the matrix.
No, the determinant of a matrix is not enough to prove its similarity to another matrix. While similar matrices have the same determinant, matrices with the same determinant are not necessarily similar.
Proving the similarity of matrices A and B is important because it allows us to understand the relationship between the two matrices. Similar matrices have the same eigenvectors and eigenvalues, which are important properties in many applications such as computing powers of matrices and solving differential equations.
Yes, it is possible for two matrices to have the same rank but not be similar. This can happen when the matrices have different eigenvalues, even though they have the same number of linearly independent rows or columns.
Yes, you can use elementary row or column operations to prove the similarity of matrices A and B. If you can transform matrix A into matrix B through elementary row or column operations, then the two matrices are similar. This is because these operations do not change the rank or the eigenvalues of a matrix.