Question

Consider an n×mmatrix Awith rank(A) <m. Is it always possible to write A=QRWhere Qis an n×mmatrix with orthonormal columns and Ris upper triangular? Explain.

Short answer

No it is not possible.

Step-by-step solution

Definition of an Orthonormal matrix.

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is. Where QT is the transpose of Q and I is the identity matrix.

For example,

$$\begin{gathered} {\mathrm{Q}}^{\mathrm{T}}\mathrm{Q}=\left(\begin{array}{c}{{\stackrel{\rightharpoonup }{\mathrm{q}}}_1}^{\mathrm{T}}\\ {{\stackrel{\rightharpoonup }{\mathrm{q}}}_2}^{\mathrm{T}}\\ {{\stackrel{\rightharpoonup }{\mathrm{q}}}_3}^{\mathrm{T}}\\ .\\ .\\ {{\stackrel{\rightharpoonup }{\mathrm{q}}}_{\mathrm{n}}}^{\mathrm{T}}\end{array}\right)\left({\stackrel{\rightharpoonup }{\mathrm{q}}}_1,{\stackrel{\rightharpoonup }{\mathrm{q}}}_2,{\stackrel{\rightharpoonup }{\mathrm{q}}}_3......{\stackrel{\rightharpoonup }{\mathrm{q}}}_{\mathrm{n}}\right) \\ \mathrm{If}m>n,\mathrm{then}, \end{gathered}$$

There is no n× mmatrix Q with orthonormal columns.

If the columns of a matrix are orthonormal, then they are linearly independent.

Hence, it is not possible to write A=QR.