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Chapter 8: Symmetric Matrices and Quadratic Forms

Expert-verified
Linear Algebra With Applications
Pages: 385 - 414
Linear Algebra With Applications

Linear Algebra With Applications

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

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Illustration

162 Questions for Chapter 8: Symmetric Matrices and Quadratic Forms

  1. Consider a quadratic formq(x)=xxAxonRnand a fixed vectorv→inRn. Is the transformation

    Found on Page 400

  2. Find singular value decompositions for the matrices listed in Exercises 7 through 14. Work with paper and pencil. In each case, draw a sketch analogous to Figure 4 in the text, showing the effect of the transformation on the unit circle, in three steps.

    Found on Page 412

  3. The function q(x→)=x→T[1224]x→is a quadratic form.

    Found on Page 413

  4. If Ais an invertible symmetric matrix, what is the relationship between the definiteness of A and A-1?

    Found on Page 400

  5. The singular values of any triangular matrix are the absolute values of its diagonal entries.

    Found on Page 413

  6. Show that a quadratic formq(x⇀)=X⇀xAx⇀of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix

    Found on Page 400

  7. LetLfrom R3to R3be the reflection about the line spanned by

    Found on Page 391

  8. Consider a symmetric3×3 matrixAwithA2=l3 . Is the linear transformation role="math" localid="1660729963116" T(x→)=Ax→necessarily the reflection about a subspace ofℝ3 ?

    Found on Page 391

  9. If the singular values of a 2×2matrixAare 3 and 4, then there must exist a unit vectoru→inR2such that||Au→||=4.

    Found on Page 413

  10. Show that the diagonal elements of a positive definite matrix A are positive.

    Found on Page 400

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