Question

a.Find all n×nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.

b.Show that the QRfactorization of an invertible n×nmatrix is unique. Hint: If,$\mathit{A}\mathbf{=}{\mathit{Q}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{1}}\mathbf{=}{\mathit{Q}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{2}}$ thenthe matrix$\mathit{A}\mathbf{=}{{\mathit{Q}}_{\mathbf{2}}}^{\mathbf{-}\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}\mathbf{=}{\mathit{Q}}_{\mathbf{2}}{{\mathit{R}}_{\mathbf{1}}}^{\mathbf{-}\mathbf{1}}$ is both orthogonal and upper triangular, with positive diagonal entries.

Short answer

1. n× nmatrices that are both orthogonal and upper triangular, with positive diagonal entries are

A =$I_n$

1. $Q_1=Q_2$andlocalid="1659498166202" $R_1=R_2$

Step-by-step solution

Consider for part (a).

If A is an orthogonal and upper triangular matrix, then$A^{-1}$is lower and upper triangular because$A^{-1}=A^T$ and it is an inverse of an upper triangular matrix. Thus,

$A^T=A^{-1}$

= A

Because A is orthogonal and diagonal with positive entries must be. So, $A=I_n$.

Consider for part (b).

Consider the theorem below.

Products and inverses of orthogonal matrices.

a.The product ABof two orthogonal n×nmatrices Aand Bis orthogonal.

b.The inverse ${\mathit{A}}^{\mathbf{-}\mathbf{1}}$ of an orthogonal n×nmatrix Ais orthogonal.

Thus, according to the above theorem${Q_2}^{-1}{Q}_1$is orthogonal and$R_2{R_1}^{-1}$is upper triangular with positive diagonal entries.

From the part (a) it has

${Q_2}^{-1}{Q}_2=R_1=R_2$

$=I_m$

So that, $Q_1=Q_2\mathrm{and}{R}_1=R_2$as claimed.

Hence,

1. n× nmatrices that are both orthogonal and upper triangular, with positive diagonal entries will be

$A=I_n$

1. $Q_1=Q_2\mathrm{and}{R}_1=R_2$and both are orthogonal and upper triangular, with positive diagonal entries.