Question

Find the orthogonal canonical reduction of the quadratic form
x2+y2 + z2 - 2xy - 2xz - 2yz. Also, find its principal axes.

02 Feb 2018
Answer :
Word Count : 540
The given quadratic form is: \[ Q(x, y, z) = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz \] ### Step 1: Represent the quadratic form in matrix notation A quadratic form can be represented as: \[ Q(x, y, z) = \mathbf{x}^T A \mathbf{x} \] where \(\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) and \(A\) is a symmetric matrix. We can express the given quadratic form in the matrix form by looking at the coefficients of the terms: \[ Q(x, y, z) = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz \] Thus, the matrix \(A\) is: \[ A = \begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{pmatrix} \] ### Step 2: Find the eigenvalues and
___ __________ _________ _____ _______ ________ __________ __________ ___ _________.
________ _________ ______ _______ _________ _____ _________ ____ __________.
___ _______ _________ ___ ______ _____ __________ _______.
_________ _____ _____ ________ ______ ________ ______ ______ _________ _______ ____ __________.
__________ _________ _________ _________ _______ _______.
_________ _________ ___ _________ _______ __________.
_________ _______ ____ __________ _________ ___ ____.
___ ________ _____ _____ ________ _________ _________ _______ _______ _____ _______ ______.
___ ___ __________ ___ _____ ____ ________.
________ _____ _____ __________ ___ ______ __________ __________ _________.
______ _____ ________ ______ _____ _______ ___ __________ __________.
____ _____ _______ ______ __________ __________ ___.
________ _______ ______ ___ _________ ______ _______ ____ ___.
____ __________ ____ __________ __________ _____ ___ _____.
____ ______ _________ _______ __________ ______ ______ _____ ______ ________ ___.
________ _________ _____ ________ _________.
_____ ___ ___ ___ _______.
____ ____ ________ _______ _______ ___ __________.
_____ _________ __________ ______ _______.
__________ ______ ________ _____ ______ __________ _____.
______ __________ ______ ________ ___ _________ _________ _______ ____ _________ ___ _________.
____ ___ ____ ___ ___ ____.
_______ _____ ________ ________ ____ ______ ___ _____ ____ __________ _____.
_______ _____ _____ _____ ___ ______ __________ ______ __________ _______.
_______ _________ _________ ____ _________ ______ ___ ______ ___ __________ ___ ____.
_________ _____ _____ ______ _______ ________ ___ ________ ___ ______ __________.
_____ _____ ___ ___ ___ ______.
______ _________ _____ ___ __________ __________ ____ __________ ________ _______.
_______ __________ ____ ________ ________ _____ __________ _________ _________.
___ ______ _________ ____ __________ ___ _________ _____ ___ _____.
_________ __________ _________ ____ _________ ___ ______ _________ _________ _________ __________ ________.
______ ____ ___ _______ ______ _________ _______ ________ _____ _____ ____.
____ ___ _________ ___ ______ ____ ____ _____ _________ ___.
__________ _____ __________ _______ __________ ______ _______ ______ ________ _______ _______ __________.
_________ ____ ____ ________ _________ ________ ____ ____.
_________ _____ _____ ____ _________ __________ __________ ____ ________ _________.
_____ _________ _________ _____ ___ _________ _________.
____ _______ ____ ____ _______ __________ ___ ___ ___ _________ ____.
________ ____ ______ _____ _________ __________ ___.
_____ ____ _____ ____ _______ __________ ________ ________.
_____ ___ __________ ____ __________ _________ ______ _____ ________ ________.
_______ ______ _____ _____ _______ __________ __________.
__________ __________ _______ _________ _____ ______ _____ _________.
____ _____ _________ _____ ____ ___ _____.
______ _______ _________ _________ ________ ________.
_____ ____ ______ _____ _______ _____ _______ _____.
________ __________ ___ ________ _________ ______ ____.
IGNOU ACADEMY
Your IGNOU journey, made easier.
  • Solved Assignments
  • Study Notes
  • Guess Papers
  • Question Papers
Call Us WhatsApp
STUDY • PREPARE • SUCCEED
Skip in 10
Related Questions

Let P(e) (x) = { p(x) ∈ R[x]| p(x) = p(−x)}

 

Find W = P(e) ∩ P3. Find a basis for W and find the dimension of W .

 

 

Complete the set S = {x3+x2+1,x2+x+1,x+1} to get a basis of P3

Find the inverse of the matrix

Image ignouacademy-com--p-orthogonal-30774

using row reduction.

Find the orthogonal canonical reduction of the quadratic form
x2+y2 + z2 - 2xy - 2xz - 2yz. Also, find its principal axes.

Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your answer.

  1. The operation ∇ defined by x∇ y = |ln(xy)| where ln x is the natural logarithm.
  2. The operation Δ defined by x Δ y = x2 + y3.

Also, for those operations which are binary operations, check whether they are associative and commutative.

Define T : R3 → R3 by

T (x, y, x) = (x + y, y, 2x − 2y + 2z).

Check that T satisfies the polynomial (x − 1)2(x − 2). Find the minimal polynomial of T .

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) R2 has infinitely many non-zero, proper vector subspaces.
ii) If T : V → W is a one-one linear transformation between two finite dimensional vector spaces V andW then T is invertible.
iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.
iv) Every unitary operator is invertible.
v) Every system of homogeneous linear equations has a non-zero solution.

IGNOU Solved Assignments Assignments Ignou Projects Project Previous Year Solved Exam Papers Notes Notes Sample Question Papers Sample Question Papers
Login
Top
📞
Call Support Instant phone assistance IGNOU MTE 2 2018 - Find the orthogonal canonical reduction of the quadratic formx2+y2 + | IGNOU Academy
→
🟢
WhatsApp Chat Fast live messaging
→
Email Us Business enquiries & support
→