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Linear Algebra With Applications
Found in: Page 403
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

Consider a quadratic form q on with symmetric matrix A, with rank A = r. Suppose that A has p positive eigenvalues, if eigenvalues are counted with their multiplicities. Show that there exists an orthogonal basis w1,...wn of Rn such that q(c1w1+....+cnwn)=cp2+....+cp2+....+cp2-cp+12-....-cr2.. Hint: Modify the approach outlined in and 65.

The diagonalizability of a quadratic form and definiteness property are used here to prove this.

See the step by step solution

Step by Step Solution

Step 1 0f 2: Given information

  • q(x) is the quadratic form, defined by symmetric matrix ARn×n , that is qx=xTAx with rank (A) = r and it has p positive eigen values.
  • As rank (A) = r , and (n - r ) eigenvalues are zero and since p of the r nonzero eigen values are positive and (r - p) eigenvalue are negative.
  • Let λ1,λ2,.....,λp be the positive eigenvalues and the negative eigenvalues are role="math" localid="1659700981027" λp+1,λp+2,.....,λr .
  • λr+1,λp+2,.....,λr are all zero eigenvalues.
  • As per theorem 8.2.2, we know that for orthonormal eigenbasis B=v1,v2,...vn and corresponding eigenvalue λ1,λ2,...,λp,λp+1,....,λr,λr+1,....,λn of A, the above quadratic form is diagonalizable as qx=λ1c12+λ2c22+...+λpcp2+λp+1cp+12+...+λrcr2+λr+1cr+12+...+λncn2, qx=λ1c12+λ2c22+...+λpcp2+λp+1cp+12+...+λrcr2 where λi=0 for i=r+1,...n
  • Here cis are the coordinates of x with respect to the eigenbasis B
  • Let us define C=w1,w2,....,wn where wi=viλi for i=1,2...r and wj=vj for j=r+1,...,n then, is orthogonal basis of as defined.

Step 2 of 2: Application


  • role="math" localid="1659702408287" =c1c1v1λ1+c2v2λ2+...+crvrλr+cr+1vr+1+...+cnvn=c1λ1v1+...+cpλpvp+cp+1λp+1vp+1+crλrvr+cr+1vr+1+cnvn=λ1c1λ12+...+λpcpλp2+λp+1cp+1λp+12+...+λrcrλr2+0.cr+12+...+0.cn2

, where λ1=0 for i=r+1,...,n

  • =c12+c22+...+cp2-cp+12-cp+22-...-cr2, where λi>0 for i=1,.... p and λj< 0 j=p+1,...r


It is proved using diagonalizability of a quadratic form and definiteness property.

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