Solve the following system of differential equations: with
, where
We are asked to solve the system numerically: [ \frac{dy}{dt} = A y, \quad y(0) = \begin{bmatrix}1\1\1\end{bmatrix}, \quad A = \begin{bmatrix}2 & -5 & -11 \ 0 & -2 & -9 \ 0 & 1 & 4\end{bmatrix}. ] We will use Euler’s method for a manual numerical solution. Let’s choose a step size (h = 0.1) for demonstration. --- The Euler formula for systems: [ y_{n+1} = y_n + h A y_n ] where (y_n = \begin{bmatrix}y_1^n \ y_2^n \ y_3^n\end{bmatrix}). Initial value: [ y_0 = \begin{bmatrix}1\1\1\end{bmatrix}. ] ______ ______ _________ ____ __________ _________ __________ ____ ______.
______ _______ ____ ________ _____ ___.
_________ __________ __________ ___ ________ _____ _________ ______ __________ ______ _______.
______ _______ ___ ____ __________ ____.
_____ _________ ___ _________ _______ _____ ____ ____ _________ _______ ________.
________ _________ ________ __________ ________ _________ ____ __________ ___ ____ ___.
_________ ___ ___ ___ ________ __________ _______ ______.
_____ ___ _________ _____ ____ _________ ______.
____ ______ ___ __________ ____ ______.
_________ ______ ____ ____ ____ _______ _________ ____ _______ _________ ______ _________.
____ _________ ___ _____ ____.
______ __________ __________ ____ ________ _________ ___ ____ _____ _________.
_____ _____ _________ ____ _________ __________.
________ ____ _____ ____ ________.
_______ ________ ____ ______ _______ _______ ________ ______ _____ ____ _________.
_______ ______ ________ ____ _____ _____ ________ ___ ____ ______ _________ _____.
______ ____ _______ _______ _______ _______.
_____ _____ _______ ________ ________ _________ ___ _______.
__________ ________ ________ _________ _______ ________ _________ ___ ________ ______.
_________ _____ ___ ________ ____ ______ __________ ________ ________ ___.
___ __________ __________ ________ ____ ________ ___ ___ ____ _____ ________.
_________ _________ ______ _____ ___ ______ __________ ___ __________ ______ ___ ______.
________ __________ _________ ____ ________ ___ ______ _________ __________ _________ ___ ______.
_______ _______ __________ __________ _______ __________ ____ ____ ___ ________.
_________ ________ __________ ___ __________ ________ _____.
________ ______ _________ ________ ________ ___.
___ _______ ______ ______ _______ ______ __________ __________.
__________ ________ _________ ___ _________ _____ ____ ________ _______.
______ _______ _____ _________ _________ ________ ________ ____ ________ _________ ____ ________.
__________ _______ ______ ________ ___ ____ ____ _____ _________ ___.
_______ ______ ___ __________ ________.
__________ _______ ___ ____ _____ ______ _______ ______ _______ _________ ______ _________.
_______ ______ __________ __________ ___ ___ _______ ____.
_________ ___ ____ _____ ___ _________ ________ _____.
__________ __________ __________ ______ ___.
_______ ____ _________ ______ ________.
______ ____ __________ ____ _________ ________ _______ _______ __________ ________ ___ ___.
______ ________ __________ ______ _________ ______.
______ ___ _______ __________ _______ __________ _____ ________ ________ _____ ________ ___.
____ _____ ________ _______ ________ __________ ___ _______ _______ __________ ___.
_____ ____.
Get Full Answer on WhatsApp
Find the SVD of the following matrices:
i)
ii)
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.
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.
Let
Find a unitary matrix U such that U^*AU is upper triangular.
Solve the following system of differential equations: with
, where
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