1.9. Submatrices, Partitioned Matrices, and Block Multiplication 9 We say a square matrix T is upper triangular if all entries below the main diagonal are zero thus tij = 0 for all i j. We have T = ⎛ ⎜ ⎜ ⎝ t11 t12 · · · t1n 0 t22 · · · t2n . . . . ... . . 0 0 · · · tnn ⎞ ⎟ ⎟ ⎠ . When not concerned with the precise entries in the positions above the main diag- onal, we write T = ⎛ ⎜ ⎜ ⎝ t11 ∗ · · · ∗ 0 t22 · · · ∗ . . . . ... . . 0 0 · · · tnn ⎞ ⎟ ⎟ ⎠ , and we abbreviate this as T = triang(t11,...,tnn). If S = triang(s11,...,snn) and T = triang(t11,...,tnn) are both upper triangular, then the product ST is also upper triangular, and it is triang(s11t11,...,snntnn). If tii = 0 for i = 1,...,n, that is, tij = 0 for all i ≥ j, then we say T is strictly upper triangular. The terms lower triangular and strictly lower triangular are defined in similar fashion: T is lower triangular if all entries above the main diagonal are zero and strictly lower triangular if all entries on or above the main diagonal are zero. When we deal with triangular matrices in this book we mainly use upper triangular the term triangular alone will be used as shorthand for “upper triangular”. 1.9. Submatrices, Partitioned Matrices, and Block Multiplication Let A be an m × n matrix. Suppose we have integers i1,...,ir and j1,...,js with 1 ≤ i1 i2 · · · ir ≤ n and 1 ≤ j1 j2 · · · js ≤ m. Then we can form an r × s matrix with ai p jq in position p, q. This is called a submatrix of A it is the submatrix formed from the entries in the rows i1,...,ir and columns j1,...,js. Putting R = {i1,...,ir} and S = {j1,...,js}, we use A[R,S] to denote this submatrix. If m = n and R = S, the submatrix A[R,R] is called a principal submatrix of A and denoted A[R]. Example 1.7. Let A = ⎛ ⎝ 1 2 3 4 5 6 7 8 9 ⎞ ⎠. The three principal submatrices of order two are A[{1,2}] = 1 2 4 5 , A[{1,3}] = 1 3 7 9 , and A[{2,3}] = 5 6 8 9 . For the sets R = {2, 3} and S = {1, 3}, we have A[R,S] = 4 6 7 9 . If R = {2, 3} and S = {1, 2, 3}, then A[R,S ] is the last two rows of A. We sometimes work with partitioned matrices, i.e., matrices which have been partitioned into blocks. Thus, suppose we have an m × n matrix A, and positive integers m1,...,mr and n1,...,ns such that r ∑ i=1 mi = m and ∑s i=1 nj = n. We partition the rows of A into r sets, consisting of the first m1 rows, then the next m2 rows, the next m3 rows, and so on. Similarly, we partition the columns into s sets,

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