Linear Algebra: transpose matrix

In the previous article, we have defined the transpose matrix. The transpose matrix of A is recorded as T (A), and we know that det (A) = det (A transpose ).

1. Transpose of Matrix Product
Matrix A and B, with T (AB) = T (B) * t ().
Matrix A1, a2... an, T (A1 * A2 *... an) = (T (an) * (T (an-1 ))*... * (T (A1 )).

2. Inverse Of the transpose matrix
The inverse matrix of A is recorded as I (a), and the unit matrix is recorded as E.
A * I (A) = E, T (A * I (a) = T (I (a) * T (A) = E, T (I () * A) = T (a) * t (I (a) = E. This indicates that the inverse matrix of the transpose is the transpose of the inverse.

3. Addition of transpose matrix
T (a + B) = T (A) + T (B ).

4. Vector transpose
Vectors are generally processed as column vectors, and the transpose of column vectors is the row vector.
If the vector is used as a matrix, the column vector is the Nx1 matrix, and the row vector is the 1xn matrix. If X is a row vector and Y is a column vector, x * Y is a 1x1 matrix, it can also be treated as a scalar.
Therefore, x.y = T (x) * Y = T (y) * X is used as the vector point multiplication.
Assume that A is a linear transformation matrix, ax. y = T (ax) * Y = T (x) * t (a) * Y = T (x) * (T (a) * y) = x. (T (a) Y ).