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Q7.4-23E

Expert-verified
Found in: Page 395

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# In Exercises 17–24, $$A$$ is an $$m \times n$$ matrix with a singular value decomposition $$A = U\Sigma {V^T}$$ , where $$U$$ is an $$m \times m$$ orthogonal matrix, $${\bf{\Sigma }}$$ is an $$m \times n$$ “diagonal” matrix with $$r$$ positive entries and no negative entries, and $$V$$ is an $$n \times n$$ orthogonal matrix. Justify each answer.23. Let $$U = \left( {{u_1}...{u_m}} \right)$$ and $$V = \left( {{v_1}...{v_n}} \right)$$ where the $${{\bf{u}}_i}$$ and $${{\bf{v}}_i}$$ are in Theorem 10. Show that $$A = {\sigma _1}{u_1}v_1^T + {\sigma _2}{u_2}v_2^T + ... + {\sigma _r}{u_r}v_r^T$$.

It is verified that $$A = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}$$.

See the step by step solution

## Step 1: Write the matrix

It is given that, $$U = \left( {{{\bf{u}}_{\bf{1}}}...{{\bf{u}}_{\bf{m}}}} \right)$$ and $$V = \left( {{{\bf{v}}_{\bf{1}}}...{{\bf{v}}_{\bf{n}}}} \right)$$

Therefore, $${V^T} = \left( {\begin{array}{*{20}{c}}{{\bf{v}}_1^T}\\{{\bf{v}}_n^T}\end{array}} \right)$$

## Step 2: Compute the matrix A from SVD:

From theorem 10, $$U\sum = \left\{ {\left( {{\sigma _1}{{\bf{u}}_1}...{\sigma _r}{{\bf{u}}_r}} \right)} \right\}$$ where $$r$$ is the rank of the matrix $$A$$, thus find $$A = U\sum {V^T}$$.

$$\begin{array}{c}A = U\sum {V^T}\\ = \left\{ {\left( {{\sigma _1}{{\bf{u}}_1}...{\sigma _r}{{\bf{u}}_r}} \right)} \right\}\left( {\begin{array}{*{20}{c}}{{\bf{v}}_1^T}\\{{\bf{v}}_n^T}\end{array}} \right)\\ = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}\end{array}$$

Hence, $$A = {\sigma _1}{{\bf{u}}_{\bf{1}}}{\bf{v}}_{\bf{1}}^{\bf{T}} + {\sigma _2}{{\bf{u}}_2}{\bf{v}}_2^{\bf{T}} + ... + {\sigma _r}{{\bf{u}}_r}{\bf{v}}_r^{\bf{T}}$$.