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Linear Algebra With Applications
Found in: Page 402
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Show that for every indefinite quadratic form q on R2, there exists an orthogonal basis w1,...,w2 of such that q(c1w1+c2w2)=c12-c22, Hint: Modify the approach outlined in

The diagonalizability of quadratic form and indefiniteness of the quadratic form are used to prove it.

See the step by step solution

Step by Step Solution

Step 1 of 2: Given information

  • Let qx1,x2 be an indefinite quadratic form which is defined by the symmetric matrix A2×2, where qx=xTAx and A is indefinite.
  • It has one positive and one negative eigenvalue.
  • As per the theorem , orthonormal eigenbasis is B={v1,v2} and its corresponding eigenvalues are role="math" localid="1659676117779" λ1>0,λ2 <0 .
  • The quadratic form is diagonalizable as q(x)=λ1c12+λ2c22.....(1)
  • The coordinates of X with respect to the eigenbasis B are ci's .

Step 2 0f 2: Application

  • Let us define C=w1,w2 , where wi=viλi, such that is the orthogonal basis of A. Now,
  • q(c1w1+c2w)=qc1V1λ1+c2V2λ2 =qc1λ1v1+c2λ2v2 =λ1c1λ12+λ2c2λ22 =c12λ1λ1+c22λ2λ2 =c12-c22λ1>0 and λ2<0


The problem is proved using diagonalizability of quadratic form and using indefiniteness of the quadratic form.

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