Linear Algebra pp 323-376

# Orthogonal Matrices and Quadratic Forms

• Belkacem Said-Houari
Chapter
Part of the Compact Textbooks in Mathematics book series (CTM)

## Abstract

In Definition  we introduced the symmetric matrices as those square matrices that are invariant under the transpose operation. That is, those matrices A in $$\mathcal{M}_{n}(\mathbb{K})$$ satisfying A = A T . This class of matrices has very important properties: for instance, they are diagonalizable (Theorem ) and if the entries of a symmetric matrix are real, then it has only real eigenvalues (Theorem ). Here we will study another class of matrices, whose inverses coincide with their transpose. These matrices are called the orthogonal matrices. In this section we restrict ourselves to the case of matrices with real entries. But all the results can be easily extended to the matrices with complex entries.

## References

1. 1.
H. Anton, C. Rorres, Elementary Linear Algebra: with Supplemental Applications, 11th edn. (Wiley, Hoboken, 2011)
2. 2.
M. Artin, Algebra, 2nd edn. (Pearson, Boston, 2011)
3. 3.
S. Axler, Linear Algebra Done Right. Undergraduate Texts in Mathematics, 2nd edn. (Springer, New York, 1997)Google Scholar
4. 4.
E.F. Beckenbach, R. Bellman, Inequalities, vol. 30 (Springer, New York, 1965)
5. 5.
F. Boschet, B. Calvo, A. Calvo, J. Doyen, Exercices d’algèbre, 1er cycle scientifique, 1er année (Librairie Armand Colin, Paris, 1971)Google Scholar
6. 6.
L. Brand, Eigenvalues of a matrix of rank k. Am. Math. Mon. 77(1), 62 (1970)Google Scholar
7. 7.
G.T. Gilbert, Positive definite matrices and Sylvester’s criterion. Am. Math. Mon. 98(1), 44–46 (1991)
8. 8.
R. Godement, Algebra (Houghton Mifflin Co., Boston, MA, 1968)
9. 9.
J. Grifone, Algèbre linéaire, 4th edn. (Cépaduès–éditions, Toulouse, 2011)
10. 10.
G.N. Hile, Entire solutions of linear elliptic equations with Laplacian principal part. Pac. J. Math 62, 127–140 (1976)
11. 11.
R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)
12. 12.
D. Kalman, J.E. White, Polynomial equations and circulant matrices. Am. Math. Mon. 108(9), 821–840 (2001)
13. 13.
P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd edn. (Academic Press, Orlando, FL, 1985)
14. 14.
S. Lang, Linear Algebra. Undergraduate Texts in Mathematics, 3rd edn. (Springer, New York, 1987)Google Scholar
15. 15.
L. Lesieur, R. Temam, J. Lefebvre, Compléments d’algèbre linéaire (Librairie Armand Colin, Paris, 1978)
16. 16.
H. Liebeck, A proof of the equality of column and row rank of a matrix. Am. Math. Mon. 73(10), 1114 (1966)Google Scholar
17. 17.
C.D. Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, Philadelphia, PA, 2000)
18. 18.
D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis. Mathematics and Its Applications (East European Series), vol. 61 (Kluwer Academic, Dordrecht, 1993)Google Scholar
19. 19.
C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)
20. 20.
J.M. Monier, Algèbre et géométrie, PC-PST-PT, 5th edn. (Dunod, Paris, 2007)Google Scholar
21. 21.
P.J. Olver, Lecture notes on numerical analysis, http://www.math.umn.edu/~olver/num.html. Accessed Sept 2016
22. 22.
F. Pécastaings, Chemins vers l’algèbre, Tome 2 (Vuibert, Paris, 1986)Google Scholar
23. 23.
M. Queysanne, Algebre, 13th edn. (Librairie Armand Colin, Paris, 1964)
24. 24.
J. Rivaud, Algèbre linéaire, Tome 1, 2nd edn. (Vuibert, Paris, 1982)
25. 25.
26. 26.
H. Roudier, Algèbre linéaire: cours et exercices, 3rd edn. (Vuibert, Paris, 2008)
27. 27.
B. Said-Houari, Differential Equations: Methods and Applications. Compact Textbook in Mathematics (Springer, Cham, 2015)Google Scholar
28. 28.
D. Serre, Matrices. Theory and Applications. Graduate Texts in Mathematics, vol. 216, 2nd edn. (Springer, New York, 2010)Google Scholar
29. 29.
G. Strang, Linear Algebra and Its Applications, 3rd edn. (Harcourt Brace Jovanovich, San Diego, 1988)
30. 30.
V. Sundarapandian, Numerical Linear Algebra (PHI Learning Pvt. Ltd., New Delhi, 2008)
31. 31.
H. Valiaho, An elementary approach to the Jordan form of a matrix. Am. Math. Mon. 93(9), 711–714 (1986)