Proving Symmetry and Rank of B=A - aXX^T | Unit Eigenvector Help

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In summary, symmetry in matrices refers to the property where a matrix is equal to its transpose. To prove symmetry, one must show that the matrix is equal to its transpose. The rank of a matrix is the maximum number of linearly independent rows or columns. To determine the rank, one can perform row operations until the matrix is in reduced row-echelon form and count the non-zero rows. Unit eigenvectors can be used to prove the rank of a matrix by comparing the number of linearly independent eigenvectors to the rank of the matrix.
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JerryKelly
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If A is a symmetric nxn mx of rank<=r>=1 and X is a unit eigenvector of A, with eigenvalue a not= 0, let B=A - aXX^T. Show that B is symmetric and that N(A) is a proper subspace of N(B). Conclude that rank B=<r-1.
i could show X is in N(B) but not in N(A). Does anyone know how I can prove it in general? Then,I could prove N(A) is a proper subspace of N(B). Thanks!
 
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I figured it out already.
 

1. What is the definition of symmetry in the context of matrices?

Symmetry in matrices refers to the property where a matrix is equal to its transpose, meaning that the elements in the matrix are mirrored along the main diagonal.

2. How do you prove symmetry of a matrix?

To prove symmetry of a matrix, you need to show that the matrix is equal to its transpose. This can be done by comparing the corresponding elements in the matrix and its transpose and showing that they are equal.

3. What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It is a measure of the dimension of the vector space spanned by the rows or columns of the matrix.

4. How do you determine the rank of a matrix?

The rank of a matrix can be determined by performing row operations on the matrix until it is in reduced row-echelon form. The number of non-zero rows in the reduced matrix is equal to the rank of the original matrix.

5. How can unit eigenvectors help with proving the rank of a matrix?

Unit eigenvectors can be used to prove the rank of a matrix by showing that the number of linearly independent eigenvectors is equal to the rank of the matrix. This can be done by finding the eigenvalues and corresponding eigenvectors of the matrix and checking if they are linearly independent.

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