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That is, the null space of can be determined by computing eigenvectors corresponding to the zero eigenvalue for . (b) Let where . Compute eigenvalues for the matrix by hand. For which values of the null-space of is trivial? (c) Determine using matlab, when and . Compute eigenvalues and corresponding eigenvectors for numerically. Does Matlab agree with you on the dimension of for each value of ? Include the script that you used and the computed eigenvalues and vectors to your solution. Hints: (a) The inclusion can be easily proven. To prove that , multiply from both sides with and interpret the result as . (c) Try out he commands help eig and format long in Matlab.

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Blake

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Step 1/1

(a) `A_(m*n` Matrix `rArrA^TA=B_(n*n)` matrix We know `P(A)=P(A^TA)`

By rank nulity theorem

`P(A)+ n (A)=n....(a)` `(A^TA)+n(A^TA)n=n...(b)`

From a and b

`rArrn(A)=n(A^TA)`

Because

`P(A^TA),<P(A).....(1)``P(A),<P(A)....(2)`

From (1) and (2)

`P(A)=P(A^TA)`(b) `A=[[1,1],[in,0],[0,in]]`

`rArrA^TA=[[1,in,0],[1,0,in]][[1,1],[in,0],[0,in]]``A^TA=[[1+in^2,0],[0,1+in^2]].``rArrA^TA ` is a scaalor matrix.`=>` Eigen value of `A^TA ` are `1+in^2,1+in^2.``rArr` And eigen vector for scalor matrix is `([1],[0])` and `([1],[0])` and `n(A^TA)` is always trival for every value of `in. ``=> And` Scalor matrix is always diagonalizable.`=>n(A^TA)` is always trivial for any `in.`

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