By C. G. Broyden
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Extra info for Basic Matrices: An Introduction to Matrix Theory and Practice
2 and its corollary. ;;; r";;; n - 1, there exists a non-null vector x that is orthogonal to every member 30 BASIC MATRICES of the set. l orthogonal to both XI and Xl and so on. Another is the existence, in certain circumstances, of an 'inverse' matrix, but before proceeding with this we first prove a simple lemma. 2 Let there exist a set of r linearly independent vectors which become, upon the addition of one further vector, linearly dependent. Then this further vector may be expressed as a unique linear combination of the r original vectors.
This gives A(X - Y) = 0 and X = Y from the linear independence of the columns of A. 5 If A is square and has linearly independent columns, and AX = 1 then XA = I. Since AX = I we have, post-multiplying by A, AXA = A or A(XA - I) = O. The result follows immediately from the linear independence of the columns of A. 5 tell us that it makes sense, if A is square and has linearly independent columns, to talk of the inverse of A without any further qualifications, since there exists a unique matrix which gives I either on pre- or post-multiplication by A.
Proof We note that the columns of A are linearly dependent if and only if there exists a vector x of 0 that is orthogonal to all rows of A. Since this orthogonality is independent of the ordering of the rows the lemma is established. 28 BASIC MATRICES 2. 2 The Unit Matrix In the field of real numbers we know that there is a number, one, that leaves any other number unchanged when multiplied by it. This number is sometimes referred to as unity, or the unit element. We now consider the possibility of the existence of a matrix that will leave any matrix unchanged when pre-multiplied by it, and examine its properties.
Basic Matrices: An Introduction to Matrix Theory and Practice by C. G. Broyden