By Melville Clark Jr., Kent F. Hansen
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Although the lengths of \xx , u 2 are unity, the lengths of w x , w 2 are both \/2. T h e vector χ in the w system is shown in Fig. 3. w2 F I G . 3. Vector χ in the nonorthogonal coordinate system w1, w2 . 12) x2 when the components are referred to the w x , w ponents are found to be χ[' = \Ζ2χλ(\\\/2) = χ'2' = (-Xl+x2)(\lV2) 2 system. 12 BIORTHOGONAL VECTORS 37 T h e component of a vector χ along some basis vector, such as w x or w 2 , is the length of the vector χ when projected on that basis vector.
Show that the above problem has no solution if λ equals an eigenvalue of A unless u is orthogonal to the eigenvector corresponding to the eigenvalue λ. 17. Prove the solution t x of Eq. 7) may be chosen orthogonal to the vector e x. 18. Show that if two matrices A and Β commute, then A and Β possess the same eigenvectors, and conversely. 19. 0 0 σ 1 0 0 1_ [b] "1 0 0 1 ί 1 ο 0 1. [c] 1 1 0 1 0 0 Ί [a] 1" 1 1 20. Show that the inverse of a nonsingular matrix A can be written A"1 = M A - 1V r , where the matrices Μ and V are constructed from the eigenvectors of A and A r , respectively, and where Λ is a diagonal matrix whose elements are the eigenvalues of A.
22) T h e inversion of Μ is always possible since Μ cannot vanish by the orthogonality and consequent independence of the . T h e result shows that a real symmetric matrix is similar to a diagonal matrix. T h e process of so transforming a matrix is called diagonalization. 10 NONSYMMETRIC MATRICES 27 I t is interesting to note that the similarity transformation used above is also an orthogonal transformation. 24) = M - 1. 25) and hence MT T h e normalized modal matrix is an orthogonal matrix. 1,10 Nonsymmetric Matrices T h e results of the previous section do not apply in full generality to a nonsymmetric matrix.