1. Find SVD - Singular Value Decomposition ...
`[[4,0],[3,-5]]`

Solution:

`A = `

`4` `0` `3` `-5`

---

Find Eigen vector for `A' * A`

`|A' * A-lamdaI|=0`

`(25-lamda)`   `-15`   `-15`   `(25-lamda)`

= 0

`:.(25-lamda) × (25-lamda) - (-15) × (-15)=0`

`:.(625-50lamda+lamda^2)-225=0`

`:.(lamda^2-50lamda+400)=0`

`:.(lamda-10)(lamda-40)=0`

`:.(lamda-10)=0 or (lamda-40)=0`

`:.lamda=10 or lamda=40`

`:.` The eigenvalues of the matrix `A' * A` are given by `lamda=10,40`





For Eigenvector-1 `(-1,1)`, Length L = `sqrt(|-1|^2+|1|^2)=1.4142`

So, normalizing gives `v_1=((-1)/(1.4142),(1)/(1.4142))=(-0.7071,0.7071)`

For Eigenvector-2 `(1,1)`, Length L = `sqrt(|1|^2+|1|^2)=1.4142`

So, normalizing gives `v_2=((1)/(1.4142),(1)/(1.4142))=(0.7071,0.7071)`

Solution
`U` is found using formula `u_i=1/sigma_i A*v_i`

`:. U = `

`-0.4472` `0.8944` `-0.8944` `-0.4472`

`:. Sigma = `

`sqrt(40)` `0` `0` `sqrt(10)`

`=`

`6.3246` `0` `0` `3.1623`

`:. V = `

`[v_1,v_2]`

`=`

`-0.7071` `0.7071` `0.7071` `0.7071`