Question

Every invertible matrix Acan be expressed as the product of an orthogonal matrix and an upper triangular matrix.

Short answer

True

Step-by-step solution

Step by step solution Step 1: Consider the theorem below.

QR factorization: Consider an n×mmatrix Mwith linearly independent columns ${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{v}}}_{\mathbf{m}}$. Then there exists an n×mmatrix Qwhose columns${\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\overrightarrow{\mathbf{u}}}_{\mathbf{m}}$ are orthonormal and an upper triangular matrix Rwith positive diagonal entries such that$\mathit{M}\mathbf{=}\mathit{Q}\mathit{R}$.

Determine whether the statement is true or false

Furthermore $r_{11}=\left|\right|{\overrightarrow{v}}_1\left|\right|,r_{jj}=\left|\right|{{\overrightarrow{v}}_j}^{\perp }\left|\right|$

($j=2,....,m$) and $r_{ij}={\overrightarrow{u}}_i.{\overrightarrow{v}}_j$

Since, the matrix is invertible hence its columns will be linearly independent and therefore, using QRdecomposition,there exists orthogonal matrix Q and an upper triangular matrix R such that$A=QR$.

Hence, the statement is true.