# QR factorization for a 4x4 tridiagonal symmetric matrix

## Homework Statement

$$\begin{bmatrix} a_{11} & a_{12} & 0 & 0\\ a_{12} & a_{22} & a_{23} & 0\\ 0 & a_{23} & a_{33} & a_{34} \\ 0 & 0 & a_{34} & a_{44} \\ \end{bmatrix} = \begin{bmatrix} q_{11} & q_{12} & q_{13} & q_{14} \\ q_{21} & q_{22} & q_{23} & q_{24} \\ q_{31} & q_{32} & q_{33} & q_{34} \\ q_{41} & q_{42} & q_{43} & q_{44} \\ \end{bmatrix} \begin{bmatrix} r_{11} & r_{12} & r_{13} & r_{14} \\ 0 & r_{22} & r_{23} & r_{24} \\ 0 & 0 & r_{33} & r_{34} \\ 0 & 0 & 0 & r_{44} \\ \end{bmatrix}$$

For the given 4x4 symmetric tridiagonal matrix A, determine which elements of its QR factorization is zero. The trick is to determine this visually.

## The Attempt at a Solution

I plugged a simple 4x4 symmetric tridagonal matrix into MATLAB and took its qr factorization and found that the top left element, ##r_{14}## of the matrix R and the bottom left 3 elements, ##q_{31}, q_{41}, q_{42}## of the matrix Q are zero. But the task was to determine this with ease and visually. Is there a trick to do this? I am not seeing it.

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andrewkirk
Homework Helper
Gold Member
If we look at the process for factorising, we see that the ##j##th entry in the top row of the R triangular matrix is proportional to ##\langle a_1,a_j\rangle##, where ##a_j## is the ##j##th column of the original matrix ##A##. For ##j=4## that inner product must be zero because the first two components of ##a_1## and the last two components of ##a_4## are zero.

If we look at the process for factorising, we see that the ##j##th entry in the top row of the R triangular matrix is proportional to ##\langle a_1,a_j\rangle##, where ##a_j## is the ##j##th column of the original matrix ##A##. For ##j=4## that inner product must be zero because the first two components of ##a_1## and the last two components of ##a_4## are zero.
Ahh thanks I see this. But what if some of the elements in the original matrix ##A## along the tridiagonal band are zero? This still seems to fit the definition of a tridiagonal matrix? But if, say, ##a_{11}## was 0, then the solution is not unique?

And also, how does your process apply to the elements of Q, and to rows 2-4 of matrix R?

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Ahh thanks I see this. But what if some of the elements in the original matrix ##A## along the tridiagonal band are zero? This still seems to fit the definition of a tridiagonal matrix? But if, say, ##a_{11}## was 0, then the solution is not unique?

And also, how does your process apply to the elements of Q, and to rows 2-4 of matrix R?
Edit: you mean ##j##th entry in the top row of the R triangular matrix is proportional to ##\langle q_1,a_j\rangle## right?

andrewkirk