Linear algebra and analysis recalls Lectures for PHD course on Numerical optimization Enrico Bertolazzi DIMS – Universit´ a di Trento November 21 – December 14, 2011 Linear algebra and analysis recalls 1 / 30
Outline Linear algebra 1 Analysis 2 The Separation Theorem and Farkas’ Lemma 3 Linear algebra and analysis recalls 2 / 30
Linear algebra Outline Linear algebra 1 Analysis 2 The Separation Theorem and Farkas’ Lemma 3 Linear algebra and analysis recalls 3 / 30
Linear algebra We always work with finite dimensional Euclidean vector spaces ❘ n , the natural number n denote the dimension of the space. Elements v ∈ ❘ n will be referred to as vectors, and we think them as composed of n real numbers stacked on top of each other, i.e., v 1 v 2 � T = � v = v 1 , v 2 , . . . , v n . . . v n v k being real numbers, and T denotes the transpose operator. Linear algebra and analysis recalls 4 / 30
Linear algebra Basic operation Basic operations defined for two vectors a , b ∈ ❘ n , and an arbitrary scalar α ∈ ❘ � T � T � � a = a 1 , a 2 , . . . , a n b = b 1 , b 2 , . . . , b n are: � T ∈ ❘ n ; 1 addition: a + b = � a 1 + b 1 , . . . , a n + b n � T ∈ ❘ n ; 2 multiplication by a scalar: α a = � α a 1 , . . . , α a n 3 scalar product between two vectors: ( a , b ) = a T b = � n k =1 a i b i ∈ ❘ . 4 A linear subspace L ⊂ ❘ n is a set with the two properties: for every a , b ∈ L it holds that a + b ∈ L ; 1 and for every α ∈ ❘ , a ∈ L it holds that α a ∈ L . 2 5 An affine subspace A ⊂ ❘ n is any set that can be represented as v + L := { v + x | x ∈ L } for some vector v ∈ ❘ n and some linear subspace L ⊂ ❘ n . Linear algebra and analysis recalls 5 / 30
Linear algebra Norm We associate a norm, or length, of a vector v ∈ ❘ n with a scalar product as: � � v � = ( v , v ) The Cauchy–Bunyakowski–Schwarz inequality says that for a , b ∈ ❘ n ( a , b ) ≤ � a � � b � we define the angle θ between two vectors via ( a , b ) cos θ = � a � � b � . We say that a is orthogonal to b if and only if ( a , b ) = 0 . The only vector orthogonal to itself is 0 = (0 , . . . , 0) T ; moreover, this is the only vector with zero norm. Linear algebra and analysis recalls 6 / 30
Linear algebra Linear and affine dependence The scalar product is symmetric and bilinear, i.e., for every a , b , c , α , β it holds that ( a , b ) = ( b , a ) , and ( α a + β b , c ) = α ( a , c ) + β ( b , c ) A collection of vectors ( v 1 , . . . , v k ) is said to be linearly independent if and only if k � α i v i = 0 ⇒ α 1 = · · · = α k = 0 . i =1 Similarly, a collection of vectors ( v 1 , . . . , v k ) is said to be affinely independent if and only if the collection ( v 2 − v 1 , v 3 − v 1 , . . . , v k − v 1 ) is linearly independent. Linear algebra and analysis recalls 7 / 30
Linear algebra Basis The largest number of linearly independent vectors in ❘ n is n ; n linearly independent vectors from ❘ n is referred to as basis. The basis ( v 1 , . . . , v n ) is said to be orthogonal if ( v i , v j ) = 0 for all i � = j . If, in addition � v i � = 1 for i = 1 , . . . , n , the basis is called orthonormal. Given the basis ( v 1 , . . . , v n ) every vector v can be written in a unique way as v = � n i =1 α i v i , and the n -tuple ( α 1 , . . . , α n ) will be referred to as coordinates of v in this basis. If the basis ( v 1 , . . . , v n ) is orthonormal, the coordinates α i are computed as α i = ( v , v i ) . The space ❘ n will be typically equipped with the standard basis ( e 1 , . . . , e n ) where e i = (0 , . . . , 0 , 1 , 0 , . . . , 0) T . For every vector v = ( v 1 , . . . , v n ) T we have ( v , e i ) = v i which allows us to identify vectors and their coordinates. Linear algebra and analysis recalls 8 / 30
Linear algebra Matrices All linear functions from ❘ n to ❘ k may be described using a linear space of real matrices ❘ k × n (i.e., with k row and n columns). Given a matrix A ∈ ❘ k × n it will often be convenient to view it as a row of its columns, which are thus vectors in ❘ k . Let A ∈ ❘ k × n have elements A ij we write A = ( a 1 , . . . , a n ) , where a i = ( A 1 i , . . . , A ki ) T ∈ ❘ k . The addition of two matrices and scalar-matrix multiplication are defined in a straightforward way. For v = ( v 1 , . . . , v n ) ∈ ❘ n we define n � v i a i ∈ ❘ k Av = i =1 Linear algebra and analysis recalls 9 / 30
Linear algebra Matrix norm and transpose We also define a norm of the matrix A by � A � = v ∈ ❘ n , � v � =1 � Av � max For a given matrix A ∈ ❘ k × n we define A T ∈ ❘ n × k with elements ( A T ) ij = A ji as matrix transpose A more elegant definition: A T is the unique matrix, satisfying the equality ( Av , u ) = ( v , A T u ) for all v ∈ ❘ n and u ∈ ❘ k . � and � � A T � From this definition it should be clear that � A � = that ( A T ) T = A Linear algebra and analysis recalls 10 / 30
Linear algebra Matrix product Given two matrices A ∈ ❘ k × n and B ∈ ❘ n × m , we define the product matrix product C = AB ∈ ❘ k × m elementwise by n � C ij = A iℓ B ℓj , i = 1 , . . . , k j = 1 , . . . , m. ℓ =1 In other words, C = AB iff for all v ∈ ❘ n , Cv = A ( Bv ) . The matrix product is: associative i.e., A ( BC ) = ( AB ) C ; not commutative i.e., AB � = BA in general; for matrices of compatible sizes. Linear algebra and analysis recalls 11 / 30
Linear algebra Matrix norm and product It is easy (and instructive) to check that � AB � ≤ � A � � B � and that ( AB ) T = B T A T . Vectors v ∈ ❘ n can be (and sometimes will be) viewed as matrices v ∈ ❘ n × 1 . Check that this embedding is norm-preserving, i.e., the norm of v viewed as a vector equals the norm of v viewed as a matrix with one column. The triangle inequality for vectors and matrices is valid � a + b � ≤ � a � + � b � , � A + B � ≤ � A � + � B � � a − b � ≥ � a � − � b � , � A − B � ≥ � A � − � B � Linear algebra and analysis recalls 12 / 30
Linear algebra Matrix inverse For a square matrix A ∈ ❘ n × n we can discuss the existence of the unique matrix A − 1 , called the inverse of A , verifying A − 1 Av = v for all v ∈ ❘ n . If the inverse of a given matrix exists, we call the latter nonsingular. The inverse matrix exists iff the columns of A are linearly independent; the columns of A T are linearly independent; the system Ax = v has a unique solution for every v ∈ ❘ n ; the system Ax = 0 has x = 0 as its unique solution. From this definition it follows that A is nonsingular iff A T is nonsingular, and, furthermore, ( A − 1 ) T = ( A T ) − 1 and therefore will be denoted simply as A − T . At last, if A and B are two nonsingular matrices of the same size, then AB is nonsingular and ( AB ) − 1 = B − 1 A − 1 . Linear algebra and analysis recalls 13 / 30
Linear algebra Eigenvalues and eigenvectors (1 / 2) If for some vector v ∈ ❘ n , and some scalar α ∈ ❘ it holds that Av = α v , we call α an eigenvalue of A and v an eigenvector, corresponding to eigenvalue α . Eigenvectors, corresponding to a given eigenvalue, form a linear subspace of ❘ n ; two nonzero eigenvectors, corresponding to two distinct eigenvalues are linearly independent. In general, every matrix A ∈ ❘ n × n has n eigenvalues (counted with multiplicity), maybe complex, which are furthermore roots of the characteristic equation det( A − λ I ) = 0 , where I ∈ ❘ n × n is the identity matrix, characterized by the fact that for all v ∈ ❘ n : Iv = v . Linear algebra and analysis recalls 14 / 30
Linear algebra Eigenvalues and eigenvectors (2 / 2) In general we have � A � ≥ | λ n | where λ n is the eigenvalue with largest absolute value. The matrix A is nonsingular iff none of its eigenvalues are equal to zero, and in this case the eigenvalues of A − 1 are equal to the reciprocal of the eigenvalues of A . The eigenvalues of A T are equal to the eigenvalues of A . We call A symmetric iff A T = A . All eigenvalues of symmetric matrices are real, and eigenvectors corresponding to distinct eigenvalues are orthogonal. Linear algebra and analysis recalls 15 / 30
Analysis Outline Linear algebra 1 Analysis 2 The Separation Theorem and Farkas’ Lemma 3 Linear algebra and analysis recalls 16 / 30
Analysis Taylor series A function f ( x ) has the expansion f ( x + h ) = f ( x ) + hf ′ ( x ) + · · · + h k k ! f ( k ) ( x ) + E where the error term E take the forms � h E = 1 ( h − t ) k f ( k +1) ( x + t ) d t, [Peano] k ! 0 h k +1 ( k + 1)! f ( k +1) ( x + η ) , = η ∈ (0 , h ) [Lagrange] = O ( h k +1 ) Linear algebra and analysis recalls 17 / 30
Analysis Multi-index notation Given a list of (non negative) integer α = ( α 1 , α 2 , . . . , α n ) called multi-index and a vector z ∈ ❘ n and a function f : ❘ n �→ ❘ we define α ! = α 1 ! α 2 ! · · · α n ! | α | = α 1 + α 2 + · · · + α n z α = z α 1 1 z α 2 2 · · · z α n n = ∂ | α | f ( z 1 , z 2 , . . . , z n ) ∂f ( z ) ∂ α ∂α 1 ∂α 2 · · · ∂α n Linear algebra and analysis recalls 18 / 30
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