4 I also borrow MATLAB notation for sets of integer labels 1 n 1 2 n It is up

4 i also borrow matlab notation for sets of integer

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4. I also borrow MATLAB notation for sets of integer labels 1 : n = { 1, 2, . . . , n } . It is up to you to review the details of matrix algebra that derive from the following basic formulas for matrix addition, multiplication, and scalar multiplication: 1. Given A = [ a ij ] i =1: m , j =1: n and B = [ b ij ] i =1: m , j =1: n (both of the same size), the sum C = A + B = [ c ij ] i =1: m , j =1: n is the matrix of the same size with components c ij = a ij + b ij , for all i , j 2. Given A = [ a ij ] i =1: m , j =1: n , B = [ b ij ] i =1: n , j =1: p , the product C = A * B = [ c ij ] i =1: m , j =1: p is the matrix of size [ m , p ] with components c ik = n j =1 a ij b jk , for all i , k We typically omit the * , and write C = AB for the matrix product. 3. Given A = [ a ij ] i =1: m , j =1: n and a complex number α , the scalar multiple C = α A is the matrix of the same size with components c ij = αa ij , for all i , j Definition 1. 1. We have zero matrices O m , n of all sizes. 2. The identity matrix I n of size [ n , n ] (a “square” matrix) has components 1 on the diagonal, and 0 off the diagonal. 3. Some, but not all, square matrices A have an inverse A - 1 such that A - 1 A = AA - 1 = I n . When the inverse exists it will be unique and A is called an invertible matrix .

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1.2. VECTOR SPACES 7 4. The transpose of A = [ a ij ] i =1: m , j =1: n is the matrix A T = [ a T ij ] i =1: n , j =1: m with a T ij = ( a ji ) for all i , j . 5. The complex conjugate of A = [ a ij ] i =1: m , j =1: n is the matrix A * = [ a * ij ] i =1: m , j =1: n with a * ij = ( a ij ) * for all i , j . 6. The Hermitian conjugate of A is the matrix A = ( A T ) * = ( A * ) T . Another very important property of matrix algebra, often overlooked in elementary treatments, is that it respects block decompositions . For example, one can think of a [4, 6] matrix as a [2, 3] matrix whose components are [2, 2] matrices. Moreover, matrix multiplication can be performed in any way that respects such a block decomposition. We will see examples of this in the following discussion. 1.2 Vector Spaces We zoom straight to complex vector spaces. Definition 2. A set of objects a , b , c , . . . called vectors is called a complex vector space V if for all vectors a , b , c , . . . and scalars α , β , γ , . . . : 1. the set is closed under a vector addition with associativity a + ( b + c ) = ( a + b ) + c and commutativity a + b = b + a . 2. the set is closed under multiplication by scalars α , β , · · · ∈ C , which is associative α ( βa ) = ( αβ ) a and distributive α ( a + b ) = αa + αb , ( α + β ) a = αa + βa . 3. there is a zero vector 0 such that 0 + a = a . 4. the scalar 1 is such that 1 a = a . 5. all vectors a have a negative - a such that a + ( - a ) = 0 . Here are some important examples of complex vector spaces, and one example of a real vector space: Examples 1. 1. “Standard” C n , which we identify as the set of complex [ n , 1] matrices X (i.e. column n -vectors). 2. For each pair m , n of positive integers, the set of complex [ m , n ] matrices. 3. C [0, 1] , the set of continuous functions from the interval [0, 1] to C . 4. For each N , including N = , W N , the set of complex trigonometric polynomials of the form f ( x ) = N n = - N a n e inx where a n C for each n .
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