Non-singular matrix and Gauss
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ENGG2013 Unit 7 Non-singular matrix and Gauss-Jordan elimination Jan, 2011. Outline • Matrix arithmetic – Matrix addition, multiplication • Non-singular matrix • Gauss-Jordan elimination kshum ENGG2013 2 The love function: a normal case Function L Domain Function L’ Range Range Boy 1 Girl A Girl A Boy 1 Boy 2 Girl B Girl B Boy 2 Boy 3 Girl C Girl C Boy 3 Boy 4 Girl D Girl D Boy 4 Boy 5 Girl E Girl E Boy 5 L(Boy 1) = Girl A, kshum Domain but L’(Girl A) = Boy 4. ENGG2013 3 The love function: a utopian case Function L Domain Function L’ Domain Range Range Boy 1 Girl A Girl A Boy 1 Boy 2 Girl B Girl B Boy 2 Boy 3 Girl C Girl C Boy 3 Boy 4 Girl D Girl D Boy 4 Boy 5 Girl E Girl E Boy 5 This function L’ is the inverse of L kshum ENGG2013 4 The love function: no inverse Function L Domain Domain Range Range Boy 1 Girl A Girl A Boy 1 Boy 2 Girl B Girl B Boy 2 Boy 3 Girl C Girl C Boy 3 Boy 4 Girl D Girl D Boy 4 Boy 5 Girl E Girl E Boy 5 This is not a function This function L has no inverse kshum ENGG2013 5 Undo-able Rotate 90 degrees clockwise Multiplied by Rotate 90 degrees counter-clockwise Multiplied by A matrix which represents a reversible process is called invertible or non-singular. kshum ENGG2013 6 Objectives • How to determine whether a matrix is invertible? • If a matrix is invertible, how to find the corresponding inverse matrix? kshum ENGG2013 7 MATRIX ALGEBRA kshum ENGG2013 8 Matrix equality • Two matrices are said to be equal if 1. They have the same number of rows and the same number of columns (i.e. same size). 2. The corresponding entry are identical. kshum ENGG2013 9 Matrix addition and scalar multiplication • We can add two matrices if they have the same size • To multiply a matrix by a real number, we just multiply all entries in the matrix by that number. kshum ENGG2013 10 Matrix multiplication • Given an mn matrix A and a pq matrix B, their product AB is defined if n=p. • If n = p, we define their product, say C = AB, by computing the (i,j)-entry in C as the dot product of the i-th row of A and the j-th row of B. m q kshum mn ENGG2013 pq 11 Examples is undefined. is undefined. kshum ENGG2013 12 Square matrix • A matrix with equal number of columns and rows is called a square matrix. • For square matrices of the same size, we can freely multiply them without worrying whether the product is well-defined or not. – Because multiplication is always well-defined in this case. • The entries with the same column and row index are called the diagonal entries. – For example: kshum ENGG2013 13 Compatibility with function composition Multiplied by Multiplied by Multiplied by kshum ENGG2013 14 Order does matter in multiplication Rotate 90 degrees Reflection around x-axis Multiplied by Multiplied by Are they the same? kshum Reflection around x-axis Rotate 90 degrees Multiplied by Multiplied by ENGG2013 15 Non-commutativity • For real numbers, we have 35 = 53. – Multiplication of real numbers is commutative. • For matrices, in general AB BA. – Multiplication of matrices is non-commutative. – For example kshum ENGG2013 16 Associativity • For real numbers, we have (34)5 = 3(45). – Multiplication of real numbers is associative. • For any three matrices A, B, C, it is always true that (AB)C = A(BC), provided that the multiplications are well-defined. – Multiplication of matrices is associative. kshum ENGG2013 17 INVERTIBLE MATRIX kshum ENGG2013 18 Identity matrix • A square matrix whose diagonal entries are all one, and off-diagonal entries are all zero, is called an identity matrix. • We usually use capital letter I for identity matrix, or add a subscript and write In if we want to stress that the size is nn. kshum ENGG2013 19 Multiplication by identity matrix is trivial • Identity matrix is like a do-nothing process. – There is no change after multiplication by the identity matrix Multiplied by • IA = A for any A. • BI = B for any B. kshum ENGG2013 20 Invertible matrix • Given an nn matrix A, if we can find a matrix A’, such that then A is said to be invertible, or non-singular. • This matrix A’ is called an inverse of A. Multiplied by Multiplied by A A’ Multiplied by In kshum ENGG2013 21 Example implies kshum Rotate 90 CW Rotate 90 CCW Multiplied by Multiplied by ENGG2013 is invertible. 22 Matrix inverse may not exist • If matrix A induces a many-to-one mapping, then we cannot hope for any inverse. has no inverse kshum ENGG2013 23 Naïve method for computing matrix inverse • Consider • Want to find A’ such that A A’= I • Solve for p, q, r, s in kshum ENGG2013 24 Uniqueness of matrix inverse • Before we discuss how to compute matrix inverse, we first show there is at most one A’ such that A A’ = A’ A = I. • Suppose on the contrary that there is another matrix A’’ such that A A’’ = A’’ A = I. • We want to prove that A’ = A’’. kshum ENGG2013 25 Proof of uniqueness Defining property of A’’ Multiply by A’ from the left I times anything is the same thing Matrix multiplication is associative Defining property of A’ I times anything is the same thing kshum ENGG2013 26 Notation • Since the matrix inverse (if exists) is unique, we use the symbol A-1 to represent the unique matrix which satisfies • We say that A-1 is the inverse of A. kshum ENGG2013 27 A convenient fact • To check that a matrix B is the inverse of A, it is sufficient to check either 1. BA = I, or 2. AB = I. • It can be proved that (1) implies (2), and (2) implies (1). – The details is left as exercise. kshum ENGG2013 28 GAUSS-JORDAN ELIMINATION kshum ENGG2013 29 Row operation using matrix • Recall that there are three kind of elementary row operations 1. Row exchange 2. Multiply a row by a non-zero constant 3. Replace a row by the sum of itself and a constant multiple of another row. • We can perform elementary row operation by matrix multiplication (from the left). • All three kinds of operation are invertible. kshum ENGG2013 30 Row exchange • Example: exchange row 2 and row 3 Multiply the same matrix from the left again, we get back the original matrix. kshum ENGG2013 31 Multiply a row by a constant • Multiply the first row by -1. Multiply the same matrix from the left again, we get back the original matrix. kshum ENGG2013 32 Row replacement • Add the first row to the second row Multiply by another matrix from the left to undo kshum ENGG2013 33 Elementary matrix (I) • Three types of elementary matrices Col. j Col. i 1. Exchange row i and row j Row i Row j kshum ENGG2013 34 Elementary matrix (II) Col. i 2. Multiply row i by m Row i kshum ENGG2013 35 Elementary matrix (III) Col. j Col. i 3. Add s times row i to row j Row i Row j kshum ENGG2013 36 Row reduction • A series of row reductions is the same as multiplying from the left a series of elementary matrices. … E1, E2, E3, … are elementary matrices. kshum ENGG2013 37 If we can row reduce to identity • Then A is non-singular, or invertible. (Matrix multiplication is associative) kshum ENGG2013 38 Gauss-Jordan elimination • It is convenient to append an identity matrix to the right • We can interpret it as If we can row reduce A to the identity by a series of row operations then we can apply the same series of row operations to I and obtain the inverse of A. kshum ENGG2013 39 Algorithm • Input: an nn matrix A. • Create an n 2n matrix M – The left half is A – The right half is In • Try to reduce the expanded matrix M such that the left half is equal to In. • If succeed, the right half of M is the inverse of A. • If you cannot reduce the left half of M to , then A is not invertible, a.k.a. singular. kshum ENGG2013 40 Example • Find the inverse of 1. Create a 36 matrix 2. After some row reductions we get • Answer: kshum ENGG2013 41
