1. Find Singular Value Decomposition (SVD) of a Matrix ...
`[[4,0],[3,-5]]`

Solution:

`A = `

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

`:. A * A' = `

`16` `12` `12` `34`

Find Eigen vector for `A * A'`

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

`(16-lamda)`   `12`   `12`   `(34-lamda)`

= 0

`:.(16-lamda) × (34-lamda) - 12 × 12=0`

`:.(544-50lamda+lamda^2)-144=0`

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

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

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

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



For Eigenvector-1 `(1/2,1)`, Length L = `sqrt(0.5^2+1^2)=1.11803`

So, normalizing gives `u_1=(0.5/1.11803,1/1.11803)=(0.4472,0.8944)`

For Eigenvector-2 `(-2,1)`, Length L = `sqrt((-2)^2+1^2)=2.23607`

So, normalizing gives `u_2=((-2)/2.23607,1/2.23607)=(-0.8944,0.4472)`

---

`:. A' * A = `

`25` `-15` `-15` `25`

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 `

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



For Eigenvector-1 `(-1,1)`, Length L = `sqrt((-1)^2+1^2)=1.41421`

So, normalizing gives `v_1=((-1)/1.41421,1/1.41421)=(-0.7071,0.7071)`

For Eigenvector-2 `(1,1)`, Length L = `sqrt(1^2+1^2)=1.41421`

So, normalizing gives `v_2=(1/1.41421,1/1.41421)=(0.7071,0.7071)`

`1^"st"` Solution

`:. Sigma = `

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

`=`

`6.32456` `0` `0` `3.16228`

`:. U = `

`[u_1,u_2]`

`=`

`0.44721` `-0.89443` `0.89443` `0.44721`

`V` is found using formula `v_i=1/sigma_i A^T*u_i`

`:. V = `

`0.70711` `-0.70711` `-0.70711` `-0.7071`

Or
`2^"nd"` Solution

`:. Sigma = `

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

`=`

`6.32456` `0` `0` `3.16228`

`:. V = `

`[v_1,v_2]`

`=`

`-0.70711` `0.70711` `0.70711` `0.70711`

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

`:. U = `

`-0.44722` `0.89443` `-0.89443` `-0.44722`