By S. A. Amitsur, D. J. Saltman, George B. Seligman
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Additional info for Algebraists' Homage: Papers in Ring Theory and Related Topics
Xn1 xm2 . . xmn For example A = A square matrix X is symmetric if X = X. ,xi j = yi j , for i = 1, 2, . . , m and j = 1, 2, . . , n. ,if xi j = 0 for i = 1, 2, . . , m and j = 1, 2, . . , n then X = 0. ,if xi j = 0 for all i = j (and xii = 0 for at least one i). ,if xii = 1 for i = 1, 2, . . , n and xi j = 0 for for all i = j for i, j = 1, 2, . . , n then X = In . It is referred to as the identity matrix. 1 0 ... 0 0 1 ... 0 In = . . .. 0 .. 0 0 ... ,the vector (xii ).
The 2 × 1 matrix B = (1, 0) has B B = 1 and BB = I2 . Beware: two conformable matrices U and V are sometimes said to be orthogonal if UV = 0. 1, Page 22). More strictly it is better to say that U is orthogonal to V; otherwise saying U and V are orthogonal could mean that both are orthogonal matrices. 3 √ − 23 − 12 −1 1 , B= 1 √2 − 23 cos(θ ) − sin(θ ) sin(θ ) cos(θ ) , B= cos(θ ) − sin(θ ) − sin(θ ) − cos(θ ) 1 A= √ 2 1 1 are both orthogonal. A= are both orthogonal for any value of θ and it can be shown that any 2 × 2 orthogonal matrix is of one of these two forms.
The vector (xii ). If u is a vector, diag(u) is the diagonal matrix with the elements of u along the diagonal and 0s elsewhere. So diag(diag(X)) is a square matrix formed by setting all off-diagonal elements of X to 0. Some texts will call diag(X) this matrix but the form diag(diag(X)) here conforms with R syntax. ,trace(X) = tr(X) = tr(xi j ) = ∑ni=1 xii . Note that tr(In ) = n. 3 Matrix Arithmetic Addition and subtraction of matrices of the same order are performed element by element (just as with vectors): X + Y = (xi j ) + (yi j ) = (xi j + yi j ), Note that X + Y = Y + X (commutativity) and (X + Y) + Z = X + (Y + Z) (associativity), provided X, Y and Z are all of the same order.
Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman