# Quadratic Forms and Matrices

## 1st Edition

### An Introductory Approach

**Authors:**N. A. Yefimov

**eBook ISBN:**9781483267678

**Imprint:**Academic Press

**Published Date:**1st January 1952

**Page Count:**174

## Description

Quadratic Forms and Matrices: An Introductory Approach focuses on the principles, processes, methodologies, and approaches involved in the study of quadratic forms and matrices.

The publication first offers information on the general theory of quadratic curves, including reduction to canonical form of the general equation of a quadratic curve, invariants and classification, reduction to canonical form of the equation of a quadratic curve with center at the origin, and transformation of coordinates in the plane. The text then examines the general theory of quadratic surfaces. Topics include transformation of rectangular coordinates in space; general deductions based on the formulas for the transformation of coordinates; reduction to canonical form of the equation of a quadric with center at the origin; and reduction to canonical form of the general equation of a quadric surface.

The manuscript ponders on linear transformations and matrices, including reduction of a quadratic form to canonical form; reduction to canonical form of the matrix of a symmetric linear transformation of space; change of the matrix of a linear transformation due to a change of basis; and geometric meaning of the determinant of a linear transformation.

The publication is a vital reference for researchers interested in the study of quadratic forms and matrices.

## Table of Contents

Preface

Translator's Note

Chapter I General Theory of Quadratic Curves

1. Transformation of Coordinates in the Plane

2. Reduction to Canonical Form of the Equation of a Quadratic Curve with Center at the Origin

3. Invariants and Classification of Quadratic Forms in Two Variables

4. Reduction to Canonical Form of the General Equation of a Quadratic Curve

5. Equation of the Center. Test for Degeneracy of a Quadratic Curve. Examples

Chapter II General Theory of Quadric Surfaces

6. Transformation of Rectangular Coordinates in Space

7. Some General Deductions Based on the Formulas for the Transformation of Coordinates

8. Reduction to Canonical Form of the Equation of a Quadric with Center at the Origin

9. Invariants and Classification of Quadratic Forms in Three Variables

10. Reduction to Canonical Form of the General Equations of a Quadric Surface

11. Equation of the Center. Test for Degeneracy of a Quadric. Examples

Chapter III Linear Transformations and Matrices

12. Linear Transformations of the Plane

13. Multiplication of Linear Transformations of the Plane and of Square Matrices of Order Two. Addition of Matrices. Multiplication of a Matrix by a Number

14. A Theorem of the Determinant of a Product of Two Matrices

15. Geometric Meaning of the Determinant of a Linear Transformation. Singular Transformations

16. The Inverse of a Linear Transformation of the Plane

17. Change of the Coordinates of a Vector Due to a Change of Basis

18. Change of the Matrix of a Linear Transformation Due to a Change of Basis

19. Matrix Form of a System of Two Linear Equations

20. Linear Transformations in Three-Dimensional Space and Square Matrices of Order Three

21. Eigenvectors of a Linear Transformation

22. Characteristic Equation of a Matrix of a Linear Transformation

23. Symmetric Linear Transformations. Reduction to Canonical Form of the Matrix of a Symmetric Linear Transformation

24. Reduction to Canonical Form of the Matrix of a Symmetric Linear Transformation of Space

25. Reduction of a Quadratic Form to Canonical Form. Application to the Theory of Quadratic Curves and Quadric Surfaces

Appendix I Vectors. Operations on Vectors and Some of Their Properties

Appendix II Elements of the Theory of Determinants

1. Determinants of Order Two and Systems of Two Linear Equations in Two Unknowns

2. Homogeneous System of Two Linear Equations in Three Unknowns

3. Determinants of Order Three

4. Cofactors and Minors

5. Systems of Three Linear Equations in Three Unknowns

Subject Index

## Details

- No. of pages:
- 174

- Language:
- English

- Copyright:
- © Academic Press 1964

- Published:
- 1st January 1952

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483267678

## About the Author

### N. A. Yefimov

Nikolay Aleksandrovich Yefimov holds the Ph.D. degree in Institute for Problems in Materials Science of National Academy of Science of Ukraine (IPMS), Kiev, Ukraine in 2000. Since 2015, he has been working as a leading researcher in IPMS. Nikolay Yefimov graduated from the Kiev Polytechnic Institute in 1992 with a specialization in "physics of metals". Nikolay Yefimov is author of more than 120 scientific publication, including monograph Quasicrystals and nano-quasicrystals are new promising materials. In: Promising materials, Vitebsk, Belarus, 2009.

### Affiliations and Expertise

Institute for Problems in Materials Science (IPMS), Kiev, Ukraine