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  1. What do Q and R stand for?

  2. Why must the diagonal entries of R be positive instead of just nonzero?

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$A=QR$, $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix. –  Mhenni Benghorbal Nov 28 '12 at 6:48
    
Thanks. I know that. I mean what names do Q and R stand for? Queer Rainbow? Quixotic Renegade? –  gondonla Nov 28 '12 at 7:02
    
See here under "computing the $QR$ decomposition" and you will find the answer for part $(2)$ of your question. –  Mhenni Benghorbal Nov 28 '12 at 7:50
    
+1 for "what do Q and R stand for". My guess was that ORthogonal starts with O and tRiangulaR ends with R, but OR is a logical operator so they call it QR. Alternatively QR are two subsequent letters and M (matrix), n (count) and P (often used in definitions here) are used. In that case you would start namefinding from LU. –  naxa Jan 30 '13 at 14:28

1 Answer 1

Here, $Q$ stands for an orthogonal matrix (one whose columns and rows are orthogonal unit vectors), and $R$ is an upper triangular matrix (entries below the main diagonal are zero).

The QR algorithm can be proved by showing that the columns of $Q$ are orthogonal and the matrix $R$ is upper triangular using the Gram-Schmidt orthonormalisation method. If $A$ is non-singular, then this process yields $R$ with a positive diagonal.

The fact that $R$ now has a positive diagonal leads to its importance in uniqueness. Given two decompositions $Q_1R_1$ and $Q_2R_2$ with upper triangular factors having positive diagonals, write $Q^*_2Q_1= R_2R^{-1}_1$. Note that $R_2R^{-1}_1$ is upper triangular and has a positive diagonal, and $Q^*_2Q_1$ is unitary. The only upper triangular unitary matrices are diagonal matrices, and the only unitary matrix with positive diagonal is the identity matrix. This leads to $Q^*_2Q=I$ and hence $Q_1=Q_2$.

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