Question

 Find the SVD of the following matrices: 



i) equation



ii) equation


 

10 Jan 2026
Answer :
Word Count : 570
i) Consider the matrix ( A = \begin{bmatrix}-1 & 1 & 1 \ 1 & 1 & 0\end{bmatrix} ). To find its SVD, we compute ( A A^T ) and ( A^T A ). [ A A^T = \begin{bmatrix}-1 & 1 & 1 \ 1 & 1 & 0\end{bmatrix} \begin{bmatrix}-1 & 1 \ 1 & 1 \ 1 & 0\end{bmatrix} = \begin{bmatrix}(-1)^2+1^2+1^2 & (-1)(1)+1(1)+1(0) \ 1(-1)+1(1)+0(1) & 1^2+1^2+0^2 \end{bmatrix} = \begin{bmatrix}3 & 0 \ 0 & 2\end{bmatrix} ] The eigenvalues of ( A A^T ) are ( \lambda_1 = 3 ) and ( \lambda_2 = 2 ), so the singular values are ( \sigma_1 = \sqrt{3} ), ( \sigma_2 = \sqrt{2} ). The eigenvectors of ( A A^T ) are ( u_1 = \begin{bmatrix}1 \ 0\end{bmatrix} ) and ( u_2 =
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Related Questions

 

 Which of the following statements are true and which are false? Give reasons for your answer.
i) If V is a finite dimensional vector space and equation is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of T is diagonal.
ii) Up to similarity, there is a unique equation matrix with minimal polynomial (x - 1)2(x - 2).
iii) If equation is the eigenvalue of a matrix A with characteristic polynomial f(x), equation and equation, then the geometric multiplicity of equation is at most k.
iv) If equation, then equation as equation.
v) If N is nilpotent, eN is also nilpotent.
vi) The sum of two normal matrices of the order n is normal.
vii) If P and Q are positive definite operators, P + Q is a positive definite operator.
viii) Generalised inverse of an equation matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any equation matrix is unique.

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