Composición de la leche

A system Ax= b of linear equations is called a complex system, or a

real systemif the entries of A,x andb are complex, or real, respectively. Note that when we say that the entries of a matrix or a vector are complex, we do not intend to rule out the possibility that they are real, but just to open up the possibility that they are not real.

3.3.1

Matrix Operations

IfAandB are real or complexM byN andN byK matrices, respec- tively, then the productC=ABis defined as theM byK matrix whose entryCmk is given by Cmk= N X n=1 AmnBnk. (3.1)

If x is an N-dimensional column vector, that is, x is an N by 1 matrix, then the productb=Axis theM-dimensional column vector with entries

bm= N

X

n=1

Amnxn. (3.2)

Ex. 3.5 Show that, for each k= 1, ..., K, Colk(C), thekth column of the matrixC=AB, is

It follows from this exercise that, for given matricesAandC, every column of C is a linear combination of the columns of A if and only if there is a third matrixB such thatC=AB.

For any N, we denote by I theN by N identity matrix with entries

In,n = 1 andIm,n = 0, for m, n = 1, ..., N and m 6= n. The size of I is

always to be inferred from the context.

The matrixA† is theconjugate transposeof the matrix A, that is, the

N byM matrix whose entries are

(A†)nm=Amn (3.3)

When the entries ofAare real,A† is just thetransposeofA, writtenAT_. Definition 3.1 A square matrixSissymmetricifST _{=S andHermitian} ifS† =S.

Definition 3.2 A square matrixS isnormalifS†S=SS†. Ex. 3.6 Let C=AB. Show thatC†=B†A†.

Ex. 3.7 LetDbe a diagonal matrix such thatDmm6=Dnnifm6=n. Show that ifBD=DB thenB is a diagonal matrix.

Ex. 3.8 Prove that, ifAB=BAfor everyN byNmatrixA, thenB=cI, for some constantc.

3.3.2

Matrix Inverses

We begin with the definition of invertibility.

Definition 3.3 A square matrixAis said to beinvertible, or to be anon- singularmatrix if there is a matrix B such that

AB=BA=I

whereI is the identity matrix of the appropriate size. There can be at most one such matrixB for a givenA. ThenB =A−1, theinverseof A.

Note that, in this definition, the matricesAand B must commute.

Proposition 3.1 The inverse of a square matrix A is unique; that is, if

AB=BA=I andAC=CA=I, thenB=C=A−1. Ex. 3.9 Prove Proposition 3.1.

The following proposition shows that invertibility follows from an ap- parently weaker condition.

Proposition 3.2 If A is square and there exist matrices B and C such thatAB=I andCA=I, thenB =C=A−1 _{andAis invertible.}

Ex. 3.10 Prove Proposition 3.2.

Later in this chapter, after we have discussed the concept of rank of a matrix, we will improve Proposition 3.2; a square matrix A is invertible if and only if there is a matrix B with AB = I, and, for any (possibly non-square)A, if there are matricesB andC with AB=I and CA=I

(where the twoImay possibly be different in size), thenAmust be square and invertible. The 2 by 2 matrixS= a b c d has an inverse S−1= 1 ad−bc d −b −c a

whenever thedeterminantofS, det(S) =ad−bcis not zero. More generally, associated with every complex square matrix is the complex number called its determinant, which is obtained from the entries of the matrix using formulas that can be found in any text on linear algebra. The significance of the determinant is that the matrix is invertible if and only if its determinant is not zero. This is of more theoretical than practical importance, since no computer can tell when a number is precisely zero. A matrixAthat is not square cannot have an inverse, but does have apseudo-inverse, which can be found using the singular-value decomposition.

Note that, ifAis invertible, thenAx= 0 can happen only whenx= 0. We shall show later, using the notion of the rank of a matrix, that the converse is also true: a square matrix A with the property that Ax = 0 only whenx= 0 must be invertible.

3.3.3

The Sherman-Morrison-Woodbury Identity

In a number of applications, stretching from linear programming to radar tracking, we are faced with the problem of computing the inverse of a slightly modified version of a matrixB, when the inverse ofB itself has already been computed. For example, when we use the simplex algorithm in linear programming, the matrixB consists of some, but not all, of the columns of a larger matrixA. At each step of the simplex algorithm, a new

Bnew is formed fromB=Boldby removing one column ofB and replacing

it with another column taken fromA.

ThenBnew differs fromB in only one column. Therefore

whereuis the column vector that equals the old column minus the new one, andvis the column of the identity matrix corresponding to the column of

Bold being altered. The inverse ofBnew can be obtained fairly easily from

the inverse ofBold using the Sherman-Morrison-Woodbury Identity:

The Sherman-Morrison-Woodbury Identity:IfvT_B−1_{u6= 1, then}

(B−uvT)−1=B−1+α−1(B−1u)(vTB−1), (3.5) where

α= 1−vTB−1u.

Ex. 3.11 Let B be invertible andvT_B−1_{u= 1. Show that B−uv}T _{is not} invertible. Show that Equation (3.5) holds, ifvT_B−1_{u6= 1.}