Maths WorkshopMaths problems with full solutions

Quadratic forms

31 problems · Symmetric matrices, quadratic forms, positive definiteness and reduction.

Search problems Level

Ex.266Definition and matrix of a quadratic form

#quadratic form #symmetric matrix #definition

Quadratic forms

Ex.557Polar form of a quadratic form

#polar form #symmetric bilinear form #quadratic form

Quadratic forms

Ex.1145Reducing a quadratic form by completing the square

#reduction #completing square #quadratic form

Quadratic forms

Ex.1146Negative definite quadratic form

#negative definite #signature #Sylvester

Quadratic forms

Ex.1147A degenerate quadratic form

#degenerate #signature #quadratic form

Quadratic forms

Ex.1148The hyperbola $x^2-y^2=1$

#hyperbola #quadratic form #geometry

Quadratic forms

Ex.1149An indefinite quadratic form

#indefinite #signature #quadratic form

Quadratic forms

Ex.1844Canonical form of a quadratic form

#canonical form #Gauss reduction #signature

Quadratic forms

Ex.1845Change of variables in a quadratic form

#change of variables #quadratic form #matrix

Quadratic forms

Ex.1846Comparison of two quadratic forms

#comparison #positive definite #inequalities

Quadratic forms

Ex.1847The cone $x^2+y^2-z^2=0$

#cone #quadratic form #signature

Quadratic forms

Ex.1848Diagonalization by congruence

#congruence #diagonalization #quadratic form

Quadratic forms

Ex.1849Reduction of a quadratic form by eigenvalues

#eigenvalues #orthogonal diagonalization #quadratic form

Quadratic forms

Ex.1850The ellipse $x^2+xy+y^2=1$

#ellipse #quadratic form #geometry

Quadratic forms

Ex.1851Gram matrix of a quadratic form

#Gram matrix #quadratic form #polar form

Quadratic forms

Ex.1852Checking positivity of a quadratic form

#positive definite #positivity #quadratic form

Quadratic forms

Ex.1853Positive definiteness of a 2×2 parameter family

#parameter #positive definite #minors

Quadratic forms

Ex.1854Reduction of a quadratic form with cross terms

#reduction #completing square #cross terms

Quadratic forms

Ex.1855Signature of a quadratic form in 3 variables

#signature #reduction #quadratic form

Quadratic forms

Ex.1856Sylvester's criterion for positive definiteness

#Sylvester criterion #positive definite #minors

Quadratic forms

Ex.2205Bounding a quadratic form by eigenvalues

#eigenvalues #Rayleigh quotient #inequalities

Quadratic forms

Ex.2206An inequality via a positive definite form

#inequality #positive definite #quadratic form

Quadratic forms

Ex.2207Sylvester's law of inertia

#inertia law #signature #invariance

Quadratic forms

Ex.2208LDL decomposition of a symmetric matrix

#LDL decomposition #Sylvester #quadratic form

Quadratic forms

Ex.2209Maximizing a quadratic form on the sphere

#maximum #sphere #eigenvalues

Quadratic forms

Ex.2210Minimizing a quadratic form on the sphere

#minimum #sphere #eigenvalues

Quadratic forms

Ex.2211Orthogonal diagonalization of a quadratic form

#orthogonal diagonalization #eigenvalues #quadratic form

Quadratic forms

Ex.2212Positive definiteness depending on a parameter

#parameter #positive definite #Sylvester

Quadratic forms

Ex.2213The Rayleigh quotient

#Rayleigh quotient #eigenvalues #optimization

Quadratic forms

Ex.2214Signature depending on a parameter

#signature #parameter #quadratic form

Quadratic forms

Ex.2215When are two quadratic forms equivalent?

#equivalence #signature #congruence

Quadratic forms