By D. E. Littlewood
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Extra info for A University Algebra: An Introduction to Classic and Modern Algebra
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.
As an example to illustrate that the procedure can fail for a multiple root, consider the matrix i—q i La oJ r The characteristic equation is A2 = 0, so that both latent roots are zero. If it could be transformed into diagonal form, then the matrix in diagonal form would be the null matrix. Clearly a matrix which is not null cannot be a transform of the null matrix. Hence no diagonal form exists. Consider the matrix r— 8 — 12 5” A = 1 5 ,-2 5 ,1 1 —24, - 42, 19J the first coefficient in the characteristic equation is 8 - 25 + 19 = 2.
Z, m, n, p], the equation o f a plane can be written L X = 0. The plane lx + my + nz + p — 0 Putting L = MATRICES 41 is clearly a parallel plane, since the two equations give p ' — p = 0, and are inconsistent, so that there is no finite point in common. Suppose that, referred to a new system o f axes, the coordinates are (x\ y\ z'). Let the plane x' = 0 have equation referred to the original axes, lxx + mxy + nxz + P i = 0, and the parallel plane x' = 1, lxx + mxy + nxz + P i = a. Then ll9 mX9 nl9 p x can clearly be chosen to make a = 1.