`A * A'`
= | | `1×1+0×0+1×1+0×0` | `1×0+0×1+1×0+0×1` | | | `0×1+1×0+0×1+1×0` | `0×0+1×1+0×0+1×1` | |
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= | | `1+0+1+0` | `0+0+0+0` | | | `0+0+0+0` | `0+1+0+1` | |
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Find Eigen vector for `A * A'`
`|A * A'-lamdaI|=0`
| `(2-lamda)` | `0` | | | `0` | `(2-lamda)` | |
| = 0 |
`:.(2-lamda) × (2-lamda) - 0 × 0=0`
`:.(4-4lamda+lamda^2)-0=0`
`:.(lamda^2-4lamda+4)=0`
`:.(lamda-2)(lamda-2)=0`
`:.(lamda-2)=0 or(lamda-2)=0 `
`:.` The eigenvalues of the matrix `A * A'` are given by `lamda=2`,
1. Eigenvectors for `lamda=2`
1. Eigenvectors for `lamda=2`
Now, reduce this matrix
The system associated with the eigenvalue `lamda=2`
`=>`
`:.` eigenvectors corresponding to the eigenvalue `lamda=2` is
Let `x_2=1,x_1=0`
Let `x_1=1,x_2=0`
For Eigenvector-1 `(0,1)`, Length L = `sqrt(0^2+1^2)=1`
So, normalizing gives `u_1=(0/1,1/1)=(0,1)`
For Eigenvector-2 `(1,0)`, Length L = `sqrt(1^2+0^2)=1`
So, normalizing gives `u_2=(1/1,0/1)=(1,0)`
`A' * A`
= | | `1×1+0×0` | `1×0+0×1` | `1×1+0×0` | `1×0+0×1` | | | `0×1+1×0` | `0×0+1×1` | `0×1+1×0` | `0×0+1×1` | | | `1×1+0×0` | `1×0+0×1` | `1×1+0×0` | `1×0+0×1` | | | `0×1+1×0` | `0×0+1×1` | `0×1+1×0` | `0×0+1×1` | |
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= | | `1+0` | `0+0` | `1+0` | `0+0` | | | `0+0` | `0+1` | `0+0` | `0+1` | | | `1+0` | `0+0` | `1+0` | `0+0` | | | `0+0` | `0+1` | `0+0` | `0+1` | |
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= | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | |
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`A' * A = ` | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | |
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Find Eigen vector for `A' * A`
`|A' * A-lamdaI|=0`
| `(1-lamda)` | `0` | `1` | `0` | | | `0` | `(1-lamda)` | `0` | `1` | | | `1` | `0` | `(1-lamda)` | `0` | | | `0` | `1` | `0` | `(1-lamda)` | |
| = 0 |
`:.(1-lamda) × |[(1-lamda),0,1],[0,(1-lamda),0],[1,0,(1-lamda)]|+0 × |[0,0,1],[1,(1-lamda),0],[0,0,(1-lamda)]|+1 × |[0,(1-lamda),1],[1,0,0],[0,1,(1-lamda)]|+0 × |[0,(1-lamda),0],[1,0,(1-lamda)],[0,1,0]|=0`
`:.(1-lamda) × (-2lamda+3lamda^2-lamda^3)+0 × (0)+1 × (2lamda-lamda^2)+0 × (0)=0`
`:.(-2lamda+5lamda^2-4lamda^3+lamda^4)+0+(2lamda-lamda^2)+0=0`
`:.(lamda^4-4lamda^3+4lamda^2)=0`
`:.lamda^2(lamda-2)(lamda-2)=0`
`:.lamda^2=0 or(lamda-2)=0 or(lamda-2)=0 `
`:.` The eigenvalues of the matrix `A' * A` are given by `lamda=0,2`,
1. Eigenvectors for `lamda=2`
1. Eigenvectors for `lamda=2`
= | | `-1` | `0` | `1` | `0` | | | `0` | `-1` | `0` | `1` | | | `1` | `0` | `-1` | `0` | | | `0` | `1` | `0` | `-1` | |
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Now, reduce this matrix
`R_1 larr R_1-:-1`
= | | `1` `1=-1-:-1` `R_1 larr R_1-:-1` | `0` `0=0-:-1` `R_1 larr R_1-:-1` | `-1` `-1=1-:-1` `R_1 larr R_1-:-1` | `0` `0=0-:-1` `R_1 larr R_1-:-1` | | | `0` | `-1` | `0` | `1` | | | `1` | `0` | `-1` | `0` | | | `0` | `1` | `0` | `-1` | |
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`R_3 larr R_3- R_1`
= | | `1` | `0` | `-1` | `0` | | | `0` | `-1` | `0` | `1` | | | `0` `0=1-1` `R_3 larr R_3- R_1` | `0` `0=0-0` `R_3 larr R_3- R_1` | `0` `0=-1--1` `R_3 larr R_3- R_1` | `0` `0=0-0` `R_3 larr R_3- R_1` | | | `0` | `1` | `0` | `-1` | |
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`R_2 larr R_2-:-1`
= | | `1` | `0` | `-1` | `0` | | | `0` `0=0-:-1` `R_2 larr R_2-:-1` | `1` `1=-1-:-1` `R_2 larr R_2-:-1` | `0` `0=0-:-1` `R_2 larr R_2-:-1` | `-1` `-1=1-:-1` `R_2 larr R_2-:-1` | | | `0` | `0` | `0` | `0` | | | `0` | `1` | `0` | `-1` | |
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`R_4 larr R_4- R_2`
= | | `1` | `0` | `-1` | `0` | | | `0` | `1` | `0` | `-1` | | | `0` | `0` | `0` | `0` | | | `0` `0=0-0` `R_4 larr R_4- R_2` | `0` `0=1-1` `R_4 larr R_4- R_2` | `0` `0=0-0` `R_4 larr R_4- R_2` | `0` `0=-1--1` `R_4 larr R_4- R_2` | |
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The system associated with the eigenvalue `lamda=2`
`(A' * A-2I)` | | = | | `1` | `0` | `-1` | `0` | | | `0` | `1` | `0` | `-1` | | | `0` | `0` | `0` | `0` | | | `0` | `0` | `0` | `0` | |
| | | = | |
`=>x_1-x_3=0,x_2-x_4=0`
`=>x_1=x_3,x_2=x_4`
`:.` eigenvectors corresponding to the eigenvalue `lamda=2` is
Let `x_4=1,x_3=0`
Let `x_3=1,x_4=0`
2. Eigenvectors for `lamda=0`
2. Eigenvectors for `lamda=0`
= | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | |
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Now, reduce this matrix
`R_3 larr R_3- R_1`
= | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `0` `0=1-1` `R_3 larr R_3- R_1` | `0` `0=0-0` `R_3 larr R_3- R_1` | `0` `0=1-1` `R_3 larr R_3- R_1` | `0` `0=0-0` `R_3 larr R_3- R_1` | | | `0` | `1` | `0` | `1` | |
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`R_4 larr R_4- R_2`
= | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `0` | `0` | `0` | `0` | | | `0` `0=0-0` `R_4 larr R_4- R_2` | `0` `0=1-1` `R_4 larr R_4- R_2` | `0` `0=0-0` `R_4 larr R_4- R_2` | `0` `0=1-1` `R_4 larr R_4- R_2` | |
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The system associated with the eigenvalue `lamda=0`
`(A' * A-0I)` | | = | | `1` | `0` | `1` | `0` | | | `0` | `1` | `0` | `1` | | | `0` | `0` | `0` | `0` | | | `0` | `0` | `0` | `0` | |
| | | = | |
`=>x_1+x_3=0,x_2+x_4=0`
`=>x_1=-x_3,x_2=-x_4`
`:.` eigenvectors corresponding to the eigenvalue `lamda=0` is
Let `x_4=1,x_3=0`
Let `x_3=1,x_4=0`
For Eigenvector-1 `(0,1,0,1)`, Length L = `sqrt(0^2+1^2+0^2+1^2)=1.41421`
So, normalizing gives `v_1=(0/1.41421,1/1.41421,0/1.41421,1/1.41421)=(0,0.70711,0,0.70711)`
For Eigenvector-2 `(1,0,1,0)`, Length L = `sqrt(1^2+0^2+1^2+0^2)=1.41421`
So, normalizing gives `v_2=(1/1.41421,0/1.41421,1/1.41421,0/1.41421)=(0.70711,0,0.70711,0)`
For Eigenvector-3 `(0,-1,0,1)`, Length L = `sqrt(0^2+(-1)^2+0^2+1^2)=1.41421`
So, normalizing gives `v_3=(0/1.41421,(-1)/1.41421,0/1.41421,1/1.41421)=(0,-0.70711,0,0.70711)`
For Eigenvector-4 `(-1,0,1,0)`, Length L = `sqrt((-1)^2+0^2+1^2+0^2)=1.41421`
So, normalizing gives `v_4=((-1)/1.41421,0/1.41421,1/1.41421,0/1.41421)=(-0.70711,0,0.70711,0)`
`1^"st"` Solution
`:. Sigma = ` | | `sqrt(2)` | `0` | `0` | `0` | | | `0` | `sqrt(2)` | `0` | `0` | |
| `=` | | `1.41421` | `0` | `0` | `0` | | | `0` | `1.41421` | `0` | `0` | |
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`V` is found using formula `v_i=1/sigma_i A^T*u_i`
`:. V = ` | | `0` | `0.70711` | `0` | `-0.70711` | | | `0.70711` | `0` | `-0.70711` | `0` | | | `0` | `0.70711` | `0` | `0.70711` | | | `0.70711` | `0` | `0.70711` | `0` | |
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Or`2^"nd"` Solution
`:. Sigma = ` | | `sqrt(2)` | `0` | `0` | `0` | | | `0` | `sqrt(2)` | `0` | `0` | |
| `=` | | `1.41421` | `0` | `0` | `0` | | | `0` | `1.41421` | `0` | `0` | |
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`:. V = ` | `[v_1,v_2,v_3,v_4]` | `=` | | `0` | `0.70711` | `0` | `-0.70711` | | | `0.70711` | `0` | `-0.70711` | `0` | | | `0` | `0.70711` | `0` | `0.70711` | | | `0.70711` | `0` | `0.70711` | `0` | |
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`U` is found using formula `u_i=1/sigma_i A*v_i`
Verify `1^"st"` Solution `A = U Sigma V^T`
`U×Sigma` | = | | × | | `1.4142` | `0` | `0` | `0` | | | `0` | `1.4142` | `0` | `0` | |
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= | | `0×1.4142+1×0` | `0×0+1×1.4142` | `0×0+1×0` | `0×0+1×0` | | | `1×1.4142+0×0` | `1×0+0×1.4142` | `1×0+0×0` | `1×0+0×0` | |
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= | | `0+0` | `0+1.4142` | `0+0` | `0+0` | | | `1.4142+0` | `0+0` | `0+0` | `0+0` | |
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= | | `0` | `1.4142` | `0` | `0` | | | `1.4142` | `0` | `0` | `0` | |
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`(U × Sigma)×(V^T)` | = | | `0` | `1.4142` | `0` | `0` | | | `1.4142` | `0` | `0` | `0` | |
| × | | `0` | `0.7071` | `0` | `0.7071` | | | `0.7071` | `0` | `0.7071` | `0` | | | `0` | `-0.7071` | `0` | `0.7071` | | | `-0.7071` | `0` | `0.7071` | `0` | |
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= | | `0×0+1.4142×0.7071+0×0+0×-0.7071` | `0×0.7071+1.4142×0+0×-0.7071+0×0` | `0×0+1.4142×0.7071+0×0+0×0.7071` | `0×0.7071+1.4142×0+0×0.7071+0×0` | | | `1.4142×0+0×0.7071+0×0+0×-0.7071` | `1.4142×0.7071+0×0+0×-0.7071+0×0` | `1.4142×0+0×0.7071+0×0+0×0.7071` | `1.4142×0.7071+0×0+0×0.7071+0×0` | |
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= | | `0+1+0+0` | `0+0+0+0` | `0+1+0+0` | `0+0+0+0` | | | `0+0+0+0` | `1+0+0+0` | `0+0+0+0` | `1+0+0+0` | |
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`1^"st"` Solution is possible.
`1^"st"` Solution is possible.Verify `2^"nd"` Solution `A = U Sigma V^T`
`U×Sigma` | = | | × | | `1.4142` | `0` | `0` | `0` | | | `0` | `1.4142` | `0` | `0` | |
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= | | `0×1.4142+1×0` | `0×0+1×1.4142` | `0×0+1×0` | `0×0+1×0` | | | `1×1.4142+0×0` | `1×0+0×1.4142` | `1×0+0×0` | `1×0+0×0` | |
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= | | `0+0` | `0+1.4142` | `0+0` | `0+0` | | | `1.4142+0` | `0+0` | `0+0` | `0+0` | |
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= | | `0` | `1.4142` | `0` | `0` | | | `1.4142` | `0` | `0` | `0` | |
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`(U × Sigma)×(V^T)` | = | | `0` | `1.4142` | `0` | `0` | | | `1.4142` | `0` | `0` | `0` | |
| × | | `0` | `0.7071` | `0` | `0.7071` | | | `0.7071` | `0` | `0.7071` | `0` | | | `0` | `-0.7071` | `0` | `0.7071` | | | `-0.7071` | `0` | `0.7071` | `0` | |
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= | | `0×0+1.4142×0.7071+0×0+0×-0.7071` | `0×0.7071+1.4142×0+0×-0.7071+0×0` | `0×0+1.4142×0.7071+0×0+0×0.7071` | `0×0.7071+1.4142×0+0×0.7071+0×0` | | | `1.4142×0+0×0.7071+0×0+0×-0.7071` | `1.4142×0.7071+0×0+0×-0.7071+0×0` | `1.4142×0+0×0.7071+0×0+0×0.7071` | `1.4142×0.7071+0×0+0×0.7071+0×0` | |
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= | | `0+1+0+0` | `0+0+0+0` | `0+1+0+0` | `0+0+0+0` | | | `0+0+0+0` | `1+0+0+0` | `0+0+0+0` | `1+0+0+0` | |
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`2^"nd"` Solution is possible.
`2^"nd"` Solution is possible.