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Ans 6 - v1 - Determinants Matrix Norms Inverse Mapping...

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Determinants, Matrix Norms, Inverse Mapping Theorem G. B. Folland The purpose of this notes is to present some useful facts about matrices and determinants and a proof of the inverse mapping theorem that is rather different from the one in Apostol. Notation: M n ( R ) denotes the set of all n × n real matrices. Determinants : If A M n ( R ), we can consider the rows of A : r 1 , . . . , r n . These are elements of R n , considered as row vectors. Conversely, given n row vectors r 1 , . . . , r n , we can stack them up into an n × n matrix A . Thus we can think of a function f on matrix space M n ( R ) as a function of n R n -valued variables or vice versa: f ( A ) ←→ f ( r 1 , . . . , r n ) . Basic Fact : There is a unique function det : M n ( R ) R (the “determinant” ) with the following three properties: i. det is a linear function of each row when the other rows are held fixed: that is, det( α a + β b , r 2 , . . . , r n ) = α det( a , r 2 , . . . , r n ) + β det( b , r 2 , . . . , r n ) , and likewise for the other rows. ii. If two rows of A are interchanged, det A is multiplied by - 1 : det( . . . , r i , . . . , r j , . . . ) = - det( . . . , r j , . . . , r i , . . . ) . iii. det( I ) = 1, where I denotes the n × n identity matrix. The uniqueness of the determinant follows from the discussion below; existence takes more work to establish. We shall not present the proof here but give the formulas. For n = 2 and n = 3 we have det a b c d = ad - bc, det a b c d e f g h i = aei - a fh + b fg - bdi + cdh - ceg. For general n , det A = σ (sgn σ ) A 1 σ (1) A 2 σ (2) · · · A ( n ) , where the sum is over all permuta- tions σ of { 1 , . . . , n } , and sgn σ is 1 or - 1 depending on whether σ is obtained by an even or odd number of interchanges of two numbers. (This formula is a computational nightmare for large n , being a sum of n ! terms, so it is of little use in practice. There are better ways to compute determinants, as we shall see shortly.) An important consequence of properties (i) and (ii) is iv. If one row of A is the zero vector, or if two rows of A are equal, then det A = 0 . Properties (i), (ii), and (iv) tell how the determinant of a matrix behaves under the elementary row operations: 1
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– Multiplying a row by a scalar multiplies the determinant by that scalar. – Interchanging two rows multiplies the determinant by - 1. – Adding a multiple of one row to another row leaves the determinant unchanged, because det( . . . , r i + α r j , . . . , r j , . . . ) = det( . . . , r i , . . . , r j , . . . ) + α det( . . . , r j , . . . , r j , . . . ) , by (i), and the last term is zero by (iv). This gives a reasonably efficient way to compute determinants. To wit, any n × n matrix A can be row-reduced either to a matrix with an all-zero row (whose determinant is 0) or to the identity matrix (whose determinant is 1). Just keep track of what happens to the determinant as you perform these row operations, and you will have calculated det A . (There are shortcuts for this procedure, but that’s another story. We’ll mostly be dealing with 2 × 2 and 3 × 3 matrices, for which one can just use the explicit formulas above.) This observation is also the key to the main theoretical significance of the determinant.
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