1 Matrix Algebra MATRIX OPERATIONS 2012 Pearson Education
1 Matrix Algebra MATRIX OPERATIONS © 2012 Pearson Education, Inc.
MATRIX MULTIPLICATION § Our goal is to represent this composite mapping as multiplication by a single matrix, denoted by AB, so that. See the figure below. § If A is , B is , and x is in , denote the columns of B by b 1, …, bp and the entries in x by x 1, …, xp. © 2012 Pearson Education, Inc. Slide 2. 1 - 2
MATRIX MULTIPLICATION § Then § By the linearity of multiplication by A, § The vector A (Bx) is a linear combination of the vectors Ab 1, …, Abp, using the entries in x as weights. § In matrix notation, this linear combination is written as. © 2012 Pearson Education, Inc. Slide 2. 1 - 3
MATRIX MULTIPLICATION § Thus multiplication by transforms x into A (Bx). § Definition: If A is an matrix, and if B is an matrix with columns b 1, …, bp, then the product AB is the matrix whose columns are Ab 1, …, Abp. § That is, § Multiplication of matrices corresponds to composition of linear transformations. © 2012 Pearson Education, Inc. Slide 2. 1 - 4
MATRIX MULTIPLICATION § Example 2: Compute AB, where and . § Solution: Write compute: © 2012 Pearson Education, Inc. , and Slide 2. 1 - 5
MATRIX MULTIPLICATION § Row—column rule for computing AB § If a product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. § If (AB)ij denotes the (i, j)-entry in AB, and if A is an matrix, then. © 2012 Pearson Education, Inc. Slide 2. 1 - 6
PROPERTIES OF MATRIX MULTIPLICATION § Theorem 2: Let A be an matrix, and let B and C have sizes for which the indicated sums and products are defined. a. (associative law of multiplication) b. (left distributive law) c. (right distributive law) d. for any scalar r e. (identity for matrix multiplication) © 2012 Pearson Education, Inc. Slide 2. 1 - 7
PROPERTIES OF MATRIX MULTIPLICATION § The definition of AB makes x. However for all § AB does not always equal BA! § The position of the factors in the product AB is emphasized by saying that A is right-multiplied by B or that B is left-multiplied by A. © 2012 Pearson Education, Inc. Slide 2. 1 - 8
PROPERTIES OF MATRIX MULTIPLICATION § If , we say that A and B commute with one another. § Warnings: 1. In general, . 2. The cancellation laws do not hold for matrix multiplication. That is, if , then it is not true in general that. 3. If a product AB is the zero matrix, you cannot conclude in general that either or. © 2012 Pearson Education, Inc. Slide 2. 1 - 9
POWERS OF A MATRIX § If A is an matrix and if k is a positive integer, then Ak denotes the product of k copies of A: § If A is nonzero and if x is in , then Akx is the result of left-multiplying x by A repeatedly k times. § If , then A 0 x should be x itself. § Thus A 0 is interpreted as the identity matrix. © 2012 Pearson Education, Inc. Slide 2. 1 - 10
THE TRANSPOSE OF A MATRIX § Given an matrix A, the transpose of A is the matrix, denoted by AT, whose columns are formed from the corresponding rows of A. Theorem 3: Let A and B denote matrices whose sizes are appropriate for the following sums and products. § § § For any scalar r, § © 2012 Pearson Education, Inc. Slide 2. 1 - 11
THE TRANSPOSE OF A MATRIX § The transpose of a product of matrices equals the product of their transposes in the reverse order. © 2012 Pearson Education, Inc. Slide 2. 1 - 12
1 Matrix Algebra THE INVERSE OF A MATRIX © 2012 Pearson Education, Inc.
MATRIX OPERATIONS § An matrix A is said to be invertible if there is an matrix C such that and where , the identity matrix. § In this case, C is an inverse of A. § In fact, C is uniquely determined by A, because if B were another inverse of A, then. § This unique inverse is denoted by , so that and. © 2012 Pearson Education, Inc. Slide 2. 2 - 14
MATRIX OPERATIONS § Theorem 5: If A is an invertible matrix, then for each b in , the equation has the unique solution. § Proof: Take any b in. § A solution exists because if is substituted for x, then. § So is a solution. § To prove that the solution is unique, show that if u is any solution, then u must be. § If , we can multiply both sides by and obtain , , and. © 2012 Pearson Education, Inc. Slide 2. 2 - 15
MATRIX OPERATIONS § Theorem 6: a. If A is an invertible matrix, then invertible and is b. If A and B are invertible matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. That is, c. If A is an invertible matrix, then so is AT, and the inverse of AT is the transpose of. That is, © 2012 Pearson Education, Inc. Slide 2. 2 - 16
MATRIX OPERATIONS § Proof: To prove statement (b), compute: § A similar calculation shows that © 2012 Pearson Education, Inc. . Slide 2. 2 - 17
ELEMENTARY MATRICES § An invertible matrix A is row equivalent to an identity matrix, and we can find by watching the row reduction of A to I. § An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix. © 2012 Pearson Education, Inc. Slide 2. 2 - 18
ELEMENTARY MATRICES § Example 1: Let , , , Compute E 1 A, E 2 A, and E 3 A, and describe how these products can be obtained by elementary row operations on A. © 2012 Pearson Education, Inc. Slide 2. 2 - 19
ELEMENTARY MATRICES § Solution: Verify that , , . § Addition of © 2012 Pearson Education, Inc. times row 1 of A to row 3 produces E 1 A. Slide 2. 2 - 20
ELEMENTARY MATRICES § An interchange of rows 1 and 2 of A produces E 2 A, and multiplication of row 3 of A by 5 produces E 3 A. § Left-multiplication by E 1 in Example 1 has the same effect on any matrix. § Since , we see that E 1 itself is produced by this same row operation on the identity. © 2012 Pearson Education, Inc. Slide 2. 2 - 21
ELEMENTARY MATRICES § Example 1 illustrates the following general fact about elementary matrices. § If an elementary row operation is performed on an matrix A, the resulting matrix can be written as EA, where the matrix E is created by performing the same row operation on Im. § Each elementary matrix E is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I. © 2012 Pearson Education, Inc. Slide 2. 2 - 22
ELEMENTARY MATRICES § Theorem 7: An matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into. § Proof: Suppose that A is invertible. § Then, since the equation has a solution for each b (Theorem 5), A has a pivot position in every row. § Because A is square, the n pivot positions must be on the diagonal, which implies that the reduced echelon form of A is In. That is, . © 2012 Pearson Education, Inc. Slide 2. 2 - 23
ELEMENTARY MATRICES § Now suppose, conversely, that. § Then, since each step of the row reduction of A corresponds to left-multiplication by an elementary matrix, there exist elementary matrices E 1, …, Ep such that. § That is, ----(1) § Since the product Ep…E 1 of invertible matrices is invertible, (1) leads to. © 2012 Pearson Education, Inc. Slide 2. 2 - 24
ALGORITHM FOR FINDING § Thus A is invertible, as it is the inverse of an invertible matrix (Theorem 6). Also, . § Then , which says that results from applying E 1, . . . , Ep successively to In. § This is the same sequence in (1) that reduced A to In. § Row reduce the augmented matrix. If A is row equivalent to I, then is row equivalent to. Otherwise, A does not have an inverse. © 2012 Pearson Education, Inc. Slide 2. 2 - 25
MATRIX OPERATIONS § Theorem 4: Let . If , then A is invertible and If , then A is not invertible. § The quantity is called the determinant of A, and we write § This theorem says that a matrix A is invertible if and only if det. © 2012 Pearson Education, Inc. Slide 2. 2 - 26
ALGORITHM FOR FINDING § Example 2: Find the inverse of the matrix , if it exists. § Solution: © 2012 Pearson Education, Inc. Slide 2. 2 - 27
ALGORITHM FOR FINDING © 2012 Pearson Education, Inc. Slide 2. 2 - 28
ALGORITHM FOR FINDING § Theorem 7 shows, since and , that A is invertible, . § Now, check the final answer. © 2012 Pearson Education, Inc. Slide 2. 2 - 29
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