1.11. Determinants 11 A = [T ]B has the block triangular form A11 A12 0 A22 , where A11 is the k ×k matrix representing the action of T on the subspace U, while A12 is k × (n − k) and A22 is (n − k) × (n − k). Now suppose U and W are invariant subspaces of T such that V = U ⊕ W. Let k = dim(U), let {u1,..., uk} be a basis for U, and let {wk+1, wk+2,..., wn} be a basis for W. Then B = {u1,..., uk, wk+1, wk+2,..., wn} is a basis for V, and A = [T ]B has the block diagonal form A11 0 0 A22 , where A11 is the k ×k matrix representing the action of T on the subspace U, while A22 is the (n − k) × (n − k) matrix representing the action of T on the subspace V. In this case we write A = A11 ⊕ A22. This can be extended in the obvious way for the case of t invariant subspaces, U1,..., Ut such that V = U1 ⊕ U2 ⊕ · · · ⊕ Ut, in which case we get a block diagonal matrix representation for T , A = ⎛ ⎜ ⎜ ⎝ A1 0 · · · 0 0 A2 · · · 0 . . . . ... . . 0 0 · · · At ⎞ ⎟ ⎟ ⎠ = A1 ⊕ A2 ⊕ · · · ⊕ At, where the block Ai represents the action of T on the invariant subspace Vi. 1.11. Determinants There are several ways to approach the definition of determinant. One can start with a formula, or with abstract properties, or take the more abstract approach through the theory of alternating, multilinear forms. Here, we merely review some basic formulas and facts. If A is an n × n matrix, then the determinant of A is (1.7) det(A) = σ (−1)σa1σ(1)a2σ(2) · · · anσ(n), where the sum is over all permutations σ of {1, 2,...,n} and the sign (−1)σ is plus one if σ is an even permutation and minus one if σ is an odd permutation. There are n! terms in this sum. Each is a signed product of n entries of A, and each product has exactly one entry from each row and from each column. Except for small values of n (such as two and three), formula (1.7) is generally a poor way to compute det(A) as it involves a sum with n! terms. When the matrix has few nonzero entries, formula (1.7) may be useful, but, usually, the eﬃcient way to compute det(A) is to reduce A to triangular form via row operations and use the following properties of the determinant function. (1) If i = j, adding a multiple of row i of A to row j of A will not change the determinant. The same holds if we add a multiple of column i to column j.

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