Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: x n.

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1 X. LINEAR ALGEBRA: THE BASICS OF MATRICES Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: y = a 1 + a 2 + a a n x n Or we could simply define x = (,,, x n ) as a vector in R n, and say that there is some other vector a = (a 1,a 2,,a n ) in R n so that we can express this as: y = a! x The dot product of two vectors in u,v!rn, also called the inner product of the vectors, is defined in the manner used above: u! v n i=1 u i v i As a note, when dealing with vectors and matrices, it is best not to use a little dot to indicate any kind of multiplication unless you specifically mean the dot product. Two vectors are said to be orthogonal if their dot product is zero. u! v = 0 does not necessarily mean that either u or v is zero in any component. For example, let u = (1,!1) and let v = (1,1). u! v = 1!(1) + 1!1 = 0, even though none of the components is zero. One graphical interpretation is that vectors are line segments: the vector x = (, ) goes from the origin to the point (, ) in R n. With this interpretation, orthogonal is the same thing as perpendicular. Incidentally, the angle! between two vectors in R n can always be calculated as: cos(!) = u v u v If the two vectors are orthogonal, then their dot product is zero, so the cosine of the angle between them is zero, so that angle must be 90 or 270 the vectors must be perpendicular. If you go with the graphical interpretation of vectors, orthogonality is the same thing as perpendicularity (in higher dimensions, as well). Now let s return to matrices. Matrices are collections of elements, arranged in rows and columns. We know how to add, transpose, invert, and multiply matrices. The important question to ask know is how to identify whether a matrix has an inverse, and if so, how to calculate the inverse. Thinking back to scalars, all numbers except for zero have a (multiplicative) inverse; namely, 1 x. Any number that has a (strictly) positive magnitude has an inverse. There is a similar concept to measure the magnitude of a matrix called its determinant. Fall 2007 math class notes, page 74

2 The determinant of an n! n matrix, denoted by A or det( A), measures the size of the area spanned in R n by vectors defined as the rows of the matrix. For a 2! 2 matrix, the determinant is defined by: (a 21,a 22! A = a 11 a 12 a 21 a 22 % A = a a ( a a The determinant can be interpreted as the area of the parallelogram in the drawing on the right. For a 3! 3 matrix A, the determinant can be calculated as: (a 11,a 12 )! a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 32 a 33 % A = a 11 a 22 a 23 a 32 a 33 ( a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 This determinant can be interpreted as the area inside the paralleloped spanned by the rows of the matrix. This lends intuition to what happens when two rows (or two columns, it turns out) of a matrix are identical, or simply multiples of one another. (a 31,a 32, (a 11,a 12, (a 21,a 22,a 23 ) In the two dimensional case, we end up with two vectors that lie on top of each other. The space that they span is a line, and a line has area of zero. In the three dimensional case, two vectors being multiples of each other means that the matrix describes a (flat) parallelogram at best, which has a volume of zero as well. In both cases the determinant is zero, which can be thought of as a matrix with no magnitude. These matrices will not be invertible. Fall 2007 math class notes, page 75

3 More generally, in order to calculate the determinant of a square matrix, we pick one row or column to expand along. Usually, people pick the top row unless something else looks much easier. Each element of this row or column is multiplied by the cofactor matrix of that element. For an n! n matrix A, the cofactor of the element a ij is the determinant of the ( n! 1) ( n! 1) matrix that results from removing the i-th row and j-th column of A, multiplied by!1! a 11! a 1 j! a 1n A = a i1! a ij! a in a n1! a nj! a nn %! A ij = (!1) i+ j a 11! a 1 j ( ) i+ j. That is:! a 1n a i1! a ij! a in a n1! a nj! a nn Where the greyed-out elements have been deleted. It is usually easiest to remember that the signs of the multiple on the determinants follow this alternating pattern: +! +!!%! +! +! +! +!! +! + Here are some notes on the useful properties of the determinant of a (square) matrix: Rule: If all the elements in a row (or column) of A are zero, then A = 0. Rule: If two rows (or two columns) of A are interchanged, the determinant changes signs but the absolute value is unchanged. Rule: If all the elements in a single row (or column) of A are multiplied by a constant c, then the determinant is multiplied by c. Rule: If two of the rows (or columns) of A are proportional, then A = 0. Rule: The value of A remains unchanged if a multiple of one row (or one column) is added to another row (or column). Rule: A = A!, AB = A B, and A + B! A + B Let s return to the interpretation of the determinant for a minute. For the most part, the determinant of a matrix is what it is it is not trying to measure anything. It is simply a number that has some useful properties. On the other hand, it is (very loosly) a measure of how orthogonal or how dissimilar are the vectors comprising Fall 2007 math class notes, page 76

4 the matrix. For vectors of the same length, the more dissimilar they are, the larger the determinant of the matrix. However, for now, we are primarily interested in determinants because they indicate whether a matrix can be inverted. Recall the case for scalars (the real numbers): a scalar can be inverted if and only if it has strictly positive magnitude, in other words, that x > 0. It is no coincidence that both determinants and absolute value use the same symbol: there is a relation between the two (but keep in mind that determinants can still be negative). Theorem: An n! n matrix A is invertible if and only if A! 0 A matrix with determinant of zero is called singular. Obviously, a nonsingular matrix is one whose determinant is nonzero, and thus is invertible. Provided that the matrix A is nonsingular, its inverse can be calculated by this formula: A!1 = 1 A A 11 A 21! A n1 % A 12 A 22 A n2 A 1n A 2n! A nn The matrix on the right hand side is known as the adjoint matrix of the matrix A, and is comprised of cofactors of the elements of A. Note though that the cofactor of the ij-th element of A is in the ji-th entry of adj( A). Though the formula for computing the inverse is unwieldy (and computationally intense!), the version of it for 2! 2 matrices is relatively usable:! A = a c b d % A(1 = 1 ad ( bc! d (b (c a % This is probably the most you ll be expected to do on your own. Let s take a detour away from matrices, back to vectors. Column vectors (as vectors usually are) are simply n! 1 matrices. We multiply vectors times each other along the same rules that we multiply matrices, and we can also multiply vectors by matrices. Consider some vectors,!r n of the form x i = ( i, i, i,, x ni ). We say that a vector x!r n is a linear combination of if there exist real numbers! 1,! 2,,! m such that: z =! 1 +! 2 +! 3 + +! m x m Fall 2007 math class notes, page 77

5 This is like a convex combination, except we no longer have the restriction that all the weights add to zero. Example:! 10 6 % is a linear combination of! 2 1 % and! 1 1 %. The vectors,!r n are called linearly dependent if there exist real numbers! 1,! 2,,! m not all equal to zero such that: 0 =! 1 +! 2 +! 3 + +! m x m Equivalently, they are linearly dependent if and only if one of the vectors can be written as a linear combination of the others. The vectors,!r n are called linearly independent if the are not linearly dependent. Note: If m > n, then the vectors,!r n are linearly dependent. In other words, you can never have more linearly independent vectors than the dimension of your space. If we have n vectors,,x n!r n, with x i = ( i, i,, x ni ), we can make a matrix with each column equal to one of these vectors (or each row equal to the transpose of a vector). The vectors,,x n are linearly independent if and only if the determinant of this matrix: [! x n ] = is not equal to zero.! 1 2! n 1 2 x 2n x n1 x n2! x nn % The maximum number of linearly independent rows (which is equal to the maximum number of linearly independent columns) of a matrix A is called the rank of A, which is denoted by rank A ( ). Rule: If A is an m! n matrix, then rank( A)! min{ m,n}. Rule: If A is an m! n matrix and Ax = 0 for some x!r n, x! 0 then rank( A) < n. Rule: If A is an n! n matrix, then A is invertible if and only if rank( A) = n. Rank is useful for describing whether matrices are invertible, but it will be important in econometrics. For instance, if you have n affine equations in m unknowns, it is Fall 2007 math class notes, page 78

6 impossible to determine the values of more than min{ m, n} unknowns, and even fewer if some of the equations are linear combinations of each other. In fact, the number of variables we can identify is the rank of the matrix formed from all the functions. Determinant and rank are two important properties of matrices. A third property is the trace of the matrix. The trace of an n! n matrix is defined as the sum of all diagonal elements. Here are some rules for traces: tr( A + B) = tr( A) + tr( B) tr( AB) = tr( BA) tr (!A) =! tr( A) tr ( A! ) = tr( A) Only a square matrix has a proper inverses, though any non-square matrix may have a generalized inverse, a left-inverse or a right-inverse. If A is an m! n matrix, the left-inverse (if it exists) is the n! m matrix L such that: LA = I n The right-inverse (if it exists) is the n! m matrix such that: AR = I n If the matrix is not square, it can have at most one of these (a right-inverse if m < n, and a left-inverse if m > n ). If it is square, they are the same if they exist. As tt turns out, a generalized inverse exists if and only if the matrix A is of full rank; that is, rank A ( ) = min{ m,n} (all the rows or columns are linearly dependent). Finally, there are some square matrices that occur frequently, and have special names and special properties. A symmetric matrix is one which is the same as its transpose, A = A!. The Hessian matrix (the matrix of second derivatives and crosspartials) of a function is always symmetric. A skew-symmetric matrix is one which is equal to the negative of its transpose, A =! A. I think I remember encountering one of these in econometrics. Idempotent matrices are one that are the same when multiplied by themselves as by the identity matrix, A 2 = AA = A. (These appear occasionally in econometrics.) A matrix is involutive if it is its own inverse, A 2 = I n, and orthogonal if it produces the identity matrix when multiplied by its transpose: A! A = I n. Fall 2007 math class notes, page 79

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