# qtyre¶

## Purpose¶

Computes the orthogonal-triangular (QR) decomposition of a matrix X and returns $$Q'Y$$ and $$R$$.

## Format¶

{ qty, r, e } = qtyre(y, x)
Parameters
• y (NxL matrix) – data

• x (NxP matrix) – data

Returns
• qty (NxL matrix) – unitary matrix

• r (KxP matrix) – upper triangular matrix. $$K = min(N,P)$$.

• e (Px1 vector) – permutation vector

## Remarks¶

Given $$X[., E]$$, where $$E$$ is a permutation vector that permutes the columns of $$X$$, there is an orthogonal matrix $$Q$$ such that $$Q'X[., E]$$ is zero below its diagonal, i.e.,

$\begin{split}Q′X[ ., E ] = \begin{bmatrix} R \\ 0 \end{bmatrix}\end{split}$

where $$R$$ is upper triangular. If we partition

$Q⁢ = \begin{bmatrix} Q_1 & Q_2 \end{bmatrix}$

where $$Q_1$$ has $$P$$ columns, then

$X[., E] = Q_1R$

is the QR decomposition of $$X[., E]$$.

If $$X$$ has rank $$P$$, then the columns of $$X$$ will not be permuted. If $$X$$ has rank $$M < P$$, then the $$M$$ linearly independent columns are permuted to the front of $$X$$ by $$E$$. Partition the permuted $$X$$ in the following way:

$X[ ., E ] = \begin{bmatrix} X_1 & X_2 \end{bmatrix}$

where $$X_1$$ is NxM and $$X_2$$ is $$N \times (P-M)$$. Further partition $$R$$ in the following way:

$\begin{split}R = \begin{bmatrix} R_{11} & R_{12} \\ 0 & 0 \end{bmatrix}\end{split}$

where $$R_{11}$$ is MxM and $$R_{12}$$ is $$M \times (P-M)$$. Then

$A = R_{11}^{-1}R_{12}$

and

$X_2 = X_1A$

that is, $$A$$ is an $$M \times (P-N)$$ matrix defining the linear combinations of $$X_2$$ with respect to $$X_1$$.

For most problems $$Q$$ or $$Q_1$$ is not it is required. Rather, we require $$Q'Y$$ or $$Q_1'Y$$ where $$Y$$ is an NxL matrix. Since $$Q$$ can be a very large matrix, qtyre() has been provided for the calculation of $$Q'Y$$ which will be a much smaller matrix. $$Q_1'Y$$ will be a submatrix of $$Q'Y$$. In particular,

$Q_1'Y = \text{qty}[1:P, .]$

and $$Q_2'Y$$ is the remaining submatrix:

$Q_2'Y = \text{qty}[P+1:N, .]$

Suppose that $$X$$ is an NxK dataset of independent variables and $$Y$$ is an Nx1 vector of dependent variables. Suppose further that $$X$$ contains linearly dependent columns, i.e., $$X$$ has rank $$M < P$$. Then define

$\begin{split}C = Q_1'Y[1:M, .]\\ A = R[1:M, 1:M]\end{split}$

and the vector (or matrix of $$L > 1$$) of least squares coefficients of the reduced, linearly independent problem is the solution of

$Ab = C$

To solve for b use qrsol():

b = qrsol(C, A);


If $$N < P$$, the factorization assumes the form:

$Q_1'X[.⁢, E] = \begin{bmatrix} R_1 & R_2 \end{bmatrix}$

where $$R_1$$ is a PxP upper triangular matrix and $$R_2$$ is $$P \times (N-P)$$. Thus $$Q$$ is a PxP matrix and $$R$$ is a PxN matrix containing $$R_1$$ and $$R_2$$. This type of factorization is useful for the solution of underdetermined systems. For the solution of

$X[.⁢, E]b = Y$

it can be shown that

b = qrsol(Q'Y, R1)|zeros(N-P,1);


qtyr.src