Find the SVD of the following matrices:
i)
ii)
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 ________ _________ _____ ______ _______.
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Use least squares method to find a quadratic polynomial that fits the following data:
(-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).
Solve the following system of differential equations: with
, where
If C and D are matrices such that
and D-1 exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs.
Check which of the following matrices is positive definite and which is positive semi-definite:
Also, find the square root of the positive definite matrix.
Find the QR decomposition of the matrix
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 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 matrix with minimal polynomial (x - 1)2(x - 2).
iii) If is the eigenvalue of a matrix A with characteristic polynomial f(x),
and
, then the geometric multiplicity of
is at most k.
iv) If , then
as
.
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 matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any matrix is unique.
Let . Find [T]B, [T]B' and P where
.