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matrix

2008-03-03 2:21 pm

Let A be an mxn matrix. If k=<m and p=<n, then a k x p matrix B is said to be a submatrix of A, if B can be obtained from A by deleting some set of m - k rows and n - p columns of A.

(For example,
if A =
1 2 3 4
5 6 7 8
9 10 11 12, then

6,

3
11,

1 2 4
5 6 8
9 10 12

are all examples of submatrices of A.)

Define the determinantal rank of A to be the largest k for which A has a k x k submatrix with nonzero determinant, Show that this is equal to the usual row or column rank of A.

回答 (1)

用戶8745

2008-03-03 9:24 pm

✔ 最佳答案

suppose the determinantal rank of A is k and the usual row and column of A is r.

Let B be the kxk submatrix of A with nonzero determinant. Suppose the columns of B correspond to the c_1, c_2, ..., c_k column of A. Since the columns of B are linearly independ, it is obvious that the corresponding columns of A are also linearly independent. Hence, r >= k.

On the other hand, suppose c_1, c_2, ..., c_r are linearly independent columns of A. let C be the matrix formed by these columns. Note that the column rank or C is r. By the fact that row rank = column rank, the row rank of C is equal to r. Hence we can find r linear independent rows of C. Let B be the matrix formed by these r rows of C. Then B is a rxr submatrix of A. Since the rows of B are linearly independent, B has nonzero determinant. Hence, k >= r.

Therefore, we conclude that k=r.

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