Properties of Inner Products in Vector Spaces

Chapter 6 INNER PRODUCT SPACES

Inner Product Inner product: Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms. = + v 0 (1) u v v u , , = = + (2) u v v w u v v u w , , , c , v u u , , c (3) = v v , 0 (4) where = v 0 if and only if

Note: v = Rn u v product dot ( Euclidean inner product for ) = u , general inner product for vector space V Note: A vector space V with an inner product is called an inner product space. ( ( ) + , , V Vector space: ) + , , , , V Inner product space:

Ex 1: (The Euclidean inner product for Rn) Show that the dot product in Rnsatisfies the four axioms of an inner product. Sol: = = u v ( , , 2 , ) , ( , , 2 , ) u u u v v v 1 1 n n = = + + + u v u v , u v u v u nv 1 1 2 2 n By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.

Ex 2: (A different inner product for Rn) Show that the function defines an inner product on R2, where and . + = v u Sol: = + = v u v u a = = u v ( , ) ( , ) u u v v 1 2 1 2 , 2 u v u v 1 1 2 2 + = u v v u ( ) , 2 2 , v u v u 1 1 2 2 1 1 2 2 = w ( ) ( , ) b 1w w 2 + = + + + u v w , ( v ) 2 ( v ) w u v w u v w 1 1 + 1 2 2 + w 2 = + v 2 2 u u w u u 1 u 1 v 1 1 2 2 2 2 = + + + ( 2 ) ( 2 ) u u u w 1 1 2 2 1 1 2 2 = + u v u w , ,

= + = + = u v u v ( ) , ( 2 ) ( ) ( 2 ) , c c c u v u v cu v cu v c 1 1 2 2 1 1 2 2 2 2 = + v v ( ) , 2 0 d v v 1 2 2 2 = + = = = = v v v , 0 2 0 0 ( ) 0 v v v v 1 2 1 2 Note: (An inner product on Rn) = + + + u v , , 0 c u v c u v c u v c 1 1 1 2 2 2 n n n i

Ex 3: (A function that is not an inner product) Show that the following function is not an inner product on R3. v = + u 2 u v u v u v 1 1 2 2 3 3 Sol: = Let v , 1 ( v ) 1 , 2 = 2 ( 2 + = v Then , 1 ( ) 1 )( ) 2 )( 1 ( ) 1 )( 6 0 Axiom 4 is not satisfied. Thus this function is not an inner product on R3.

Thm : (Properties of inner products) Let u, v, and w be vectors in an inner product space V, and let c be any real number. (1) = , = u 0 u u v v + v , c v 0 , , = 0 , w = + (2) w v w , (3) u v , c Norm (length) of u: = u u u || || , Note: 2= u u u || || ,

Distance between u and v: = = u v u v u v u v ( , ) || || , d Angle between two nonzero vectors u and v: u v , = cos , 0 u v || || || || Orthogonal: u v ( ) = u v , 0 u and v are orthogonal if .

Notes: (1) If , then v is called a unit vector. 1 || || = v (2) v 1 v (the unit vector in the direction of v) Normalizin g v v 0 not a unit vector

Ex 6: (Finding inner product) , p q = + + a b + 0 0 a b 1 1 ab is an inner product n n = , = q + 2 2 Let ( a ( ) 1 q 2 = , ( ( ) ) 4 || 2 = polynomial be ( ) p d c s in ? ( ) p x x ? q x b x x P x 2 = || ? ( , ) q ) p Sol: = 0 ( + ) 2 + ) 1 )( 2 = ( ) , 1 ( ) 4 )( )( ( 2 a p q = = x + ) 2 + = 2 2 2 ( ( ) ) || || p , 3 = 4 3 ( 1 21 b c q q q + = q 2 2 x = ( , ) || || , d p q p q p q p q = ) 3 + + ) 3 = 2 2 2 ( 2 ( 22

Properties of norm: u || || 0 (1) (2) if and only if 0 || || = u = u = 0 u u || || | || | c || c (3) Properties of distance: u v ( , ) 0 d (1) (2) if and only if 0 ) , ( = v u d v u d d = u = v v u ( , ) ( , ) (3)

Thm Let u and v be vectors in an inner product space V. (1) Cauchy-Schwarz inequality: u v u v | , | || || || || Theorem 5.4 (2) Triangle inequality: + + u v u v || || || || || || Theorem 5.5 (3) Pythagorean theorem : u and v are orthogonal if and only if 2 2 2 + = + u v u v || || || || || || Theorem 5.6

Example 7 For the vectors u = (1, 0) and v = (0, 1) in R2 with the Euclidean inner product find <u,v> , norm of u , the distant between u and v and the angle between them? <u,v> = 0+0=0

Example 8 If u = U and v = V are matrices in the vector space Mnn For example in M22 Find <u,v> and a norm of u

Example 9 Suppose the vectors u and v are matrices in the vector space M22 Find <u,v> , and

Example 10 If p and q are polynomials in Pn Find <p,q> and the norm of a polynomial p relative to this inner product

Example 11 Find the standard inner product on P2 of the given polynomials and the norm of p . -8-21=-29

ORTHONORMAL BASES; GRAMSCHMIDT PROCESS DEFINITION A set of vectors in an inner product space is called an orthogonal set if all pairs of distinct vectors in the set are orthogonal. An orthogonal set in which each vector has norm 1 is called orthonormal. EXAMPLE Using the Gram Schmidt Process

so an orthonormal basis for R3is

Chapter 7 Eigenvalues, Eigenvectors

7.1EIGENVALUES AND EIGENVECTORS DEFINITION

This is called the characteristic equation of A The determinant is always a polynomial p in p( ), called the characteristic polynomial of A.

EXAMPLE 3 Eigenvalues of an Upper Triangular Matrix Find the eigenvalues of the upper triangular matrix

THEOREM 7.1.1 If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are the entries on the main diagonal of A. EXAMPLE 4 Eigenvalues of a Lower Triangular Matrix By inspection, the eigenvalues of the lower triangular matrix

THEOREM 7.1.2

EXAMPLE 5 Eigenvectors and Bases for Eigenspaces Find bases for the eigenspaces of are linearly independent, these vectors form a basis for the eigenspace corresponding to =2. If =1, then

is a basis for the eigenspace corresponding to =1.

Powers of a Matrix THEOREM 7.1.3 EXAMPLE 6 : Find eigenvalues of A7

Eigenvalues and Invertibility THEOREM 7.1.4 A square matrix A is invertible if and only if =0 is not an eigenvalue of A. EXAMPLE 7 Using Theorem 7.1.4 The matrix A in Example 5 is invertible since it has eigenvalues

THEOREM 7.1.5 Equivalent Statements If A is an nxn matrix, and if TA: Rn (a) A is invertible. (b) AX=0 has only the trivial solution. (c) The reduced row-echelon form of A is In. (d) A is expressible as a product of elementary matrices. (e) AX=b is consistent for every nx1 matrix b . (f) AX=b has exactly one solution for every nx1 matrix b. Rnis multiplication by A, then the following are equivalent.

7.2DIAGONALIZATION The Matrix Diagonalization Problem DEFINITION A square matrix A is called diagonalizable if there is an invertible matrix P such that P-1AP is a diagonal matrix; the matrix P is said to diagonalize A. THEOREM 7.2.1 If A is an nxn matrix, then the following are equivalent. (a) A is diagonalizable. (b) A has n linearly independent eigenvectors.

EXAMPLE 1 Finding a Matrix P That Diagonalizes a Matrix A Find a matrix P that diagonalizes

EXAMPLE 2 A Matrix That Is Not Diagonalizable Find a matrix P that diagonalizes

The matrix A is not diagonalizable.

THEOREM 7.2.2 THEOREM 7.2.3 If an nxn matrix A has n distinct eigenvalues, then A is diagonalizable. EXAMPLE 3 Using Theorem 7.2.3

THEOREM 7.2.4 Geometric and Algebraic Multiplicity If A is a square matrix, then (a) For every eigenvalue of A, the geometric multiplicity is less than or equal to the algebraic multiplicity. (b) A is diagonalizable if and only if, for every eigenvalue, the geometric multiplicity is equal to the algebraic multiplicity. Computing Powers of a Matrix

EXAMPLE 6 Power of a Matrix