Proving/Disproving: RQ is Tridiagonal Matrix

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In summary, a tridiagonal matrix is a square matrix with non-zero elements only on the main diagonal, the diagonal above the main diagonal, and the diagonal below the main diagonal. To prove that a matrix is tridiagonal, all non-zero elements must fall within these three diagonals. A matrix can be both tridiagonal and diagonal, meaning all non-zero elements are on the main diagonal and the adjacent diagonals. To disprove that a matrix is tridiagonal, it must have at least one non-zero element outside of these three diagonals. Non-square matrices cannot be tridiagonal as they do not have the required structure.
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kat18
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Hi, I'm having some difficulty with the following, please help.

Suppose A in R^{nxn} (n>3) is a tridiagonal matrix where Q is an orthogonal matrix and R is an upper-triangular matrix such that A=QR .

Must RQ be a tridiagonal matrix?
If yes, give a proof; otherwise, construct a counterexample.
 
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RQ = Q-1AQ = QTAQ. I don't know how much that helps, but it may be a start.
 

1. What is a tridiagonal matrix?

A tridiagonal matrix is a special type of square matrix where the only non-zero elements are on the main diagonal, the diagonal above the main diagonal, and the diagonal below the main diagonal. All other elements are zero.

2. How do you prove that a matrix is tridiagonal?

To prove that a matrix is tridiagonal, you need to show that all the non-zero elements are only on the main diagonal, the diagonal above the main diagonal, and the diagonal below the main diagonal. This can be done by examining the matrix or by using mathematical techniques such as row reduction or induction.

3. Can a matrix be both tridiagonal and diagonal?

Yes, a matrix can be both tridiagonal and diagonal. This means that all the non-zero elements are on the main diagonal, as well as the two diagonals adjacent to the main diagonal.

4. How do you disprove that a matrix is tridiagonal?

If a matrix is not tridiagonal, then it must have at least one non-zero element in a position that is not on the main diagonal, the diagonal above the main diagonal, or the diagonal below the main diagonal. This can be shown by examining the matrix or using mathematical techniques such as row reduction or induction.

5. Is it possible for a non-square matrix to be tridiagonal?

No, a non-square matrix cannot be tridiagonal. Tridiagonal matrices are defined as square matrices, meaning that they have the same number of rows and columns. Therefore, a non-square matrix cannot have the required structure to be considered tridiagonal.

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