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Example text

B p ) = ( A b l t A b 2 , . . , Ab„). 48) To further illustrate matrix multiplication in terms of rows and columns, let A — be a 2 x p matrix, x be a p x 1 vector, and S b e a p x p matrix. 49) jl ) . 50) a a2 Any matrix can be multiplied by its transpose. 52)]. 54)]. 27), it is clear that both A A ' and A ' A are symmetric. In the above illustration for A B in terms of row and column vectors, the rows of A were denoted by Ά[ and the columns of B by bj. If both rows and columns of a matrix A are under discussion, as in A A ' and A'A, we will use the notation a^ for rows and a^·) for columns.

After finding λχ, λ2, · ■ ■, λ η , the accompanying eigenvectors x 1 ; x 2 , . . 105). 62) that k and A — AI commute, we obtain (A - AI)fcx = k0 = 0. 106) Thus if x is an eigenvector of A, A;x is also an eigenvector, and eigenvectors are unique only up to multiplication by a scalar. Hence we can adjust the length of x, but the direction from the origin is unique; that is, the relative values of (ratios of) the components of x = (x\, X2, ■ ■ ■, xn)' a r e unique. Typically, the eigenvector x is scaled so that x'x = 1.

C) = cl. Then n |D| = |d| = IJc = cn. 84) to any square matrix A is |cA| = c"|A|. 85) Because the determinant is a scalar, we can carry out operations such as IAI2, iAr, jJL provided that |A| > 0 for \K\ll2 and that |A| φ 0 for 1/|A|. If the square matrix A is singular, its determinant is 0: |A| = 0 if A is singular. 86) If A is near singular, then there exists a linear combination of the columns that is close to 0, and | A| is also close to 0. If A is nonsingular, its determinant is nonzero: |A| φ 0 if A is nonsingular.

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