Q29E Show that an SVD聽A=U危VTcan be ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q29E (page 413)

Show that an SVD $A = U 危 V^{T}$
can be written as $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 鈥� + 蟽_{r} {\overset{鈫�}{u}}_{r} {\overset{鈫�}{v}}_{r}^{T} .$

Short Answer

Expert verified

Use the structure of $U a n d 危$ and also, use the definition of matrix multiplication.

Step by step solution

To find that an SVDA=UΣVT

Let A be an $n 脳 m$ matrix whose rank is r. Also let the singular valued e composition of A is given by $A = U 危 V^{t}$ , where U is an orthogonal $n 脳 m$ matrix and V is an orthogonal matrix. Also $危$ is an $n 脳 m$ matrix whose first r diagonal entries are the non zero singular values of $蟽_{1}, 蟽_{2}, 鈥�, 蟽_{r}$ and all other entries are zero.

Now let, $u_{i} a n d v_{i}$ are the columns of U and V respectively. Then

$U 危 = [\begin{matrix} 鈭� & 鈭� & 鈭� & 鈭� \\ u_{1} & 鈰� & u_{r} & 0 & 鈰� & 0 \\ 鈭� & 鈭� & 鈭� & 鈭� \end{matrix}] [\begin{matrix} 蟽_{1} & 0 \\ 鈰� & 蟽_{r} \\ 0 & 0 \end{matrix}] = [\begin{matrix} 鈭� & 鈭� & 鈭� & 鈭� \\ 蟽_{1} u_{1} & 鈰� & 蟽_{r} u_{r} & 0 & 鈰� & 0 \\ 鈭� & 鈭� & 鈭� & 鈭� \end{matrix}]$

Also, we have

$V = [\begin{matrix} v_{1} & v_{2} & 鈥� & v_{m} \end{matrix}]$

can be written as A=σ1u→1v→1T+…+σru→rv→rT..

Therefore,
$A = U 危 V^{t} = [\begin{matrix} 鈭� & 鈭� & 鈭� & 鈭� \\ 蟽_{1} u_{1} & 蟽_{r} u_{r} & 0 & 0 \\ 鈭� & 鈭� & 鈭� & 鈭� \end{matrix}] [\begin{matrix} {v_{1}}^{t} \\ {v_{2}}^{t} \\ 鈰� \\ {v_{m}}^{t} \end{matrix}] = 蟽_{1} u_{1} v_{1}^{t} + 鈰� + 蟽_{r} u_{r} v_{r}^{t} + 0 = 蟽_{1} u_{1} v_{1}^{t} + 鈰� + 蟽_{r} u_{r} {v_{r}}^{t}$

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91影视

Show that an SVD $A = U 危 V^{T}$
can be written as $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 鈥� + 蟽_{r} {\overset{鈫�}{u}}_{r} {\overset{鈫�}{v}}_{r}^{T} .$

Short Answer

Step by step solution

To find that an SVDA=UΣVT

can be written as A=σ1u→1v→1T+…+σru→rv→rT..

One App. One Place for Learning.

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91影视

Show that an SVDA=U危VTcan be written asA=蟽1u鈫�1v鈫�1T+鈥�+蟽ru鈫�rv鈫�rT.

Short Answer

Step by step solution

To find that an SVDA=UΣVT

can be written as A=σ1u→1v→1T+…+σru→rv→rT..

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Logic and Functions

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Statistics

Study anywhere. Anytime. Across all devices.

Show that an SVD $A = U 危 V^{T}$
can be written as $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 鈥� + 蟽_{r} {\overset{鈫�}{u}}_{r} {\overset{鈫�}{v}}_{r}^{T} .$