# Quadratic Forms and Matrices

## An Introductory Approach

- 1st Edition - January 1, 1952
- Author: N. A. Yefimov
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 7 6 7 - 8

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

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Request a sales quoteQuadratic 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.

PrefaceTranslator's NoteChapter 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. ExamplesChapter 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. ExamplesChapter 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 SurfacesAppendix I Vectors. Operations on Vectors and Some of Their PropertiesAppendix 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 UnknownsSubject Index

- No. of pages: 174
- Language: English
- Edition: 1
- Published: January 1, 1952
- Imprint: Academic Press
- eBook ISBN: 9781483267678

NY

### 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, UkraineRead

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