Read e-book online A University Algebra: An Introduction to Classic and Modern PDF

By D. E. Littlewood

ISBN-10: 0486627152

ISBN-13: 9780486627151

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Extra info for A University Algebra: An Introduction to Classic and Modern Algebra

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I f the quadratic form is tion is to replace the matrix QUADRATIC FORMS 49 The matrix A is a symmetric matrix, which is defined as a matrix which is equal to its transpose A = A. Symmetric matrices have the following fundamental property. Theorem IV . The latent roots of a real symmetric matrix are all real. Let A be a real symmetric matrix and A and X a latent root and the corresponding pole. Then A X = XX. Take the complex conjugate. Transposing, since A = A , Thence Since A is real, this gives AX = XX.

XA — AX. X A X = AXX, since A X = AX, and X A X — AXX, since XA — AX. Since X X is a positive non-zero expression, it follows that A = A and the latent root is real. It follows immediately that the pole X can also be chosen so that it is real. Theorem V. Every real symmetric matrix can he transformed into a diagonal matrix by means of an orthogonal matrix. The theorem will be proved first for the case when all the latent roots are distinct. The procedure adopted in this case is the general procedure that should be employed for a specific example.

Or Hence the characteristic equation is A3 - 2A2 - A + 2 = 0, (A - 1)(A + 1)(A — 2) = 0. Corresponding to A — 1, the singular matrix A 7, - 12, 5" 15, - 26, 11 _ 24, - 42, 18_ 1 is The right factor of zero of this matrix is given by the equations lx — 12y + 5z = 0, 15® - 26y + llz = 0, 24® — 4%y + 18z = 0, whence, solving x = y = z. MATRICES 39 The right factor of zero is which is the pole corresponding to the latent root A = 1. Similarly for A = — 1, the right factor o f zero of “ 9, - 12, 5” 15, - 24, 11 24, - 42, 20_ - Lastly, for A = 2, Thence “ 6, - 12, 5 “ "1 ~ 3 15, - 27, 11 24, - 42, 17_ _ 6 8, - 12, 5 “ 1 .

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A University Algebra: An Introduction to Classic and Modern Algebra by D. E. Littlewood

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