Linear Algebra Lecture 14 - Quadratic Forms Oskar Kędzierski 27 January 2020 Quadratic Form Definition A function Q : Rn R is called a quadratic form if ÝÑ 2 2 Q x1,..., xn a11x1 . annxn aij xi xj , pp qq “ ` ` ` 1ďiăjďn ÿ that is, it is a function given by a homogeneous polynomial of degree 2 in variables x1,..., xn. Quadratic Form Definition A function Q : Rn R is called a quadratic form if ÝÑ 2 2 Q x1,..., xn a11x1 . annxn aij xi xj , pp qq “ ` ` ` 1ďiăjďn ÿ that is, it is a function given by a homogeneous polynomial of degree 2 in variables x1,..., xn. Examples 2 2 Q x1, x2 x1 x2 pp qq “ ´ Quadratic Form Definition A function Q : Rn R is called a quadratic form if ÝÑ 2 2 Q x1,..., xn a11x1 . annxn aij xi xj , pp qq “ ` ` ` 1ďiăjďn ÿ that is, it is a function given by a homogeneous polynomial of degree 2 in variables x1,..., xn. Examples 2 2 Q x1, x2 x1 x2 pp qq “ ´ 2 2 2 Q x1, x2, x3 x1 2x2 5x3 2x1x2 2x1x3 2x2x3 pp qq “ ` ` ` ´ ` Symmetric Matrix Recall Definition ⊺ Matrix A aij M n n; R is called symmetric if A A, i.e. “ r s P p ˆ q “ aij aji for i, j 1,..., n. “ “ Symmetric Matrix Recall Definition ⊺ Matrix A aij M n n; R is called symmetric if A A, i.e. “ r s P p ˆ q “ aij aji for i, j 1,..., n. “ “ Example 0 2 5 matrix 2 4 3 is symmetric » ´ fi 5 3 1 ´ – fl Symmetric Matrix Recall Definition ⊺ Matrix A aij M n n; R is called symmetric if A A, i.e. “ r s P p ˆ q “ aij aji for i, j 1,..., n. “ “ Example 0 2 5 matrix 2 4 3 is symmetric » ´ fi 5 3 1 ´ – fl 0 2 6 matrix 2 4 3 is not symmetric » ´ fi 5 3 1 ´ – fl Matrix of a Quadratic Form Definition n 2 Let Q x1,..., xn 1 aii x 1 aij xi xj be a quadratic pp qq “ i“ i ` ďiăjďn form. The matrix of the quadratic form Q is a symmetric matrix ř ř 1 M bij M n n; R such that bii aii and bij bji aij for “ r s P p ˆ q “ “ “ 2 1 i j n. ď ă ď Matrix of a Quadratic Form Definition n 2 Let Q x1,..., xn 1 aii x 1 aij xi xj be a quadratic pp qq “ i“ i ` ďiăjďn form. The matrix of the quadratic form Q is a symmetric matrix ř ř 1 M bij M n n; R such that bii aii and bij bji aij for “ r s P p ˆ q “ “ “ 2 1 i j n. ď ă ď Example 2 2 1 0 The matrix of the form Q x1, x2 x1 x2 is M . pp qq “ ´ “ 0 1 „ ´ Matrix of a Quadratic Form Definition n 2 Let Q x1,..., xn 1 aii x 1 aij xi xj be a quadratic pp qq “ i“ i ` ďiăjďn form. The matrix of the quadratic form Q is a symmetric matrix ř ř 1 M bij M n n; R such that bii aii and bij bji aij for “ r s P p ˆ q “ “ “ 2 1 i j n. ď ă ď Example 2 2 1 0 The matrix of the form Q x1, x2 x1 x2 is M . pp qq “ ´ “ 0 1 2 2 „2 ´ The matrix of the form Q x1, x2, x3 x1 2x2 5x3 2x1x2 pp qq “ ` ` ` 4x1x3 8x2x3 is ´ ` 1 1 2 ´ M 1 2 4 “ » fi 2 4 5 ´ – fl Matrix of a Quadratic Form (continued) Proposition Let M be a matrix of the quadratic form Q : Rn R. Then ÝÑ ⊺ Q x1,..., xn x Mx, pp qq “ x1 where x . “ » . fi x — n ffi – fl Matrix of a Quadratic Form (continued) Proposition Let M be a matrix of the quadratic form Q : Rn R. Then ÝÑ ⊺ Q x1,..., xn x Mx, pp qq “ x1 where x . “ » . fi x — n ffi – fl Proof. Entries of matrix M in the i-th row get multiplied by xi and elements in the j-th column get multiplied by xj . For every i j ‰ the monomial xi xj comes from the entry in the i-th row, j-th column and from the entry in the j-th row, i-th column. Positive/Negative Definite Quadratic Form Definition Quadratic form Q : Rn R (resp. symmetric matrix ÝÑ M M n n; R ) is positive definite if Q x 0 (resp. P p ˆ q p qą x ⊺Mx 0) for any x Rn, x 0. ą P ‰ Positive/Negative Definite Quadratic Form Definition Quadratic form Q : Rn R (resp. symmetric matrix ÝÑ M M n n; R ) is positive definite if Q x 0 (resp. P p ˆ q p qą x ⊺Mx 0) for any x Rn, x 0. ą P ‰ Quadratic form Q (resp. symmetric matrix M) is negative definite if Q x 0 (resp. x ⊺Mx 0) for any x Rn, x 0. p qă ă P ‰ Positive/Negative Definite Quadratic Form Definition Quadratic form Q : Rn R (resp. symmetric matrix ÝÑ M M n n; R ) is positive definite if Q x 0 (resp. P p ˆ q p qą x ⊺Mx 0) for any x Rn, x 0. ą P ‰ Quadratic form Q (resp. symmetric matrix M) is negative definite if Q x 0 (resp. x ⊺Mx 0) for any x Rn, x 0. p qă ă P ‰ Example The quadratic form 2 : Rn R is positive definite since 2 2 2}¨} ÝÑ x x1 . x 0 for x 0. } } “ ` ` n ą ‰ Positive/Negative Definite Quadratic Form Definition Quadratic form Q : Rn R (resp. symmetric matrix ÝÑ M M n n; R ) is positive definite if Q x 0 (resp. P p ˆ q p qą x ⊺Mx 0) for any x Rn, x 0. ą P ‰ Quadratic form Q (resp. symmetric matrix M) is negative definite if Q x 0 (resp. x ⊺Mx 0) for any x Rn, x 0. p qă ă P ‰ Example The quadratic form 2 : Rn R is positive definite since 2 2 2}¨} ÝÑ x x1 . xn 0 for x 0. } } “ ` ` ą ‰ 2 2 The quadratic form Q x1, x2 x1 x2 is not positive definite pp qq “ ´ since Q 0, 1 1 0. It is not negative definite since pp qq “´ ă Q 1, 0 1 0. pp qq “ ą Positive/Negative Definite Quadratic Form Definition Quadratic form Q : Rn R (resp. symmetric matrix ÝÑ M M n n; R ) is positive definite if Q x 0 (resp. P p ˆ q p qą x ⊺Mx 0) for any x Rn, x 0. ą P ‰ Quadratic form Q (resp. symmetric matrix M) is negative definite if Q x 0 (resp. x ⊺Mx 0) for any x Rn, x 0. p qă ă P ‰ Example The quadratic form 2 : Rn R is positive definite since 2 2 2}¨} ÝÑ x x1 . xn 0 for x 0. } } “ ` ` ą ‰ 2 2 The quadratic form Q x1, x2 x1 x2 is not positive definite pp qq “ ´ since Q 0, 1 1 0. It is not negative definite since pp qq “´ ă Q 1, 0 1 0. pp qq “ ą 2 2 2 The quadratic form Q x1, x2, x3 x1 2x2 5x3 2x1x2 pp qq2 “ ` `2 ` 2x1x3 2x2x3 x1 x2 x3 x2 2x3 is not positive ´ ` “ p ` ´ q ` p ` q definite since Q 3, 2, 1 0. It is not negative definite. pp ´ qq “ Recall 2 2 2 2 a1 a2 . an a1 a2 . a 2a1a2 2a1a3 . 2a1an p ` ` ` q “ ` ` ` n ` ` ` ` ` 2a2a3 2a2a4 . 2a2an 2a3a4 . 2a3an ...... 2an 1an ` ` ` ` ` ` ` ` ` ´ Recall 2 2 2 2 a1 a2 . an a1 a2 . a 2a1a2 2a1a3 . 2a1an p ` ` ` q “ ` ` ` n ` ` ` ` ` 2a2a3 2a2a4 . 2a2an 2a3a4 . 2a3an ...... 2an 1an ` ` ` ` ` ` ` ` ` ´ For example 2 x1 3x2 2x3 p ´ ` q “ Recall 2 2 2 2 a1 a2 . an a1 a2 . a 2a1a2 2a1a3 . 2a1an p ` ` ` q “ ` ` ` n ` ` ` ` ` 2a2a3 2a2a4 . 2a2an 2a3a4 . 2a3an ...... 2an 1an ` ` ` ` ` ` ` ` ` ´ For example 2 x1 3x2 2x3 p ´ ` q “ 2 2 2 2 2 x1 3 x2 2 x3 2 3 x1x2 2 2x1x3 2 3 2x2x3 “ ` p´ q 2` 2 ` ¨ p2´ q ` ¨ ` ¨ p´ q ¨ “ x1 9x2 4x3 6x1x2 4x1x3 12x2x3 “ ` ` ´ ` ´ Recall 2 2 2 2 a1 a2 . an a1 a2 . a 2a1a2 2a1a3 . 2a1an p ` ` ` q “ ` ` ` n ` ` ` ` ` 2a2a3 2a2a4 . 2a2an 2a3a4 . 2a3an ...... 2an 1an ` ` ` ` ` ` ` ` ` ´ For example 2 x1 3x2 2x3 p ´ ` q “ 2 2 2 2 2 x1 3 x2 2 x3 2 3 x1x2 2 2x1x3 2 3 2x2x3 “ ` p´ q 2` 2 ` ¨ p2´ q ` ¨ ` ¨ p´ q ¨ “ x1 9x2 4x3 6x1x2 4x1x3 12x2x3 “ ` ` ´ ` ´ Proposition A quadratic form Q : Rn R can expressed (possibly after a ÝÑ 2 2 2 change of coordinates) as Q x1,..., xn l1 l2 . l pp qq“˘ ˘ ˘ ˘ n where l1,..., ln are linear functions such that li ,..., ln do not depend on the variables x1,..., xi 1 for i 2,..., n. ´ “ Recall 2 2 2 2 a1 a2 . an a1 a2 . a 2a1a2 2a1a3 . 2a1an p ` ` ` q “ ` ` ` n ` ` ` ` ` 2a2a3 2a2a4 . 2a2an 2a3a4 . 2a3an ...... 2an 1an ` ` ` ` ` ` ` ` ` ´ For example 2 x1 3x2 2x3 p ´ ` q “ 2 2 2 2 2 x1 3 x2 2 x3 2 3 x1x2 2 2x1x3 2 3 2x2x3 “ ` p´ q 2` 2 ` ¨ p2´ q ` ¨ ` ¨ p´ q ¨ “ x1 9x2 4x3 6x1x2 4x1x3 12x2x3 “ ` ` ´ ` ´ Proposition A quadratic form Q : Rn R can expressed (possibly after a ÝÑ 2 2 2 change of coordinates) as Q x1,..., xn l1 l2 .