show that the length of Ax equals the length of A^T x if AA^T=A^T A
5.9
To show that the length of Ax equals the length of A^T x when AA^T=A^T A, we need to use basic properties of vector and matrix operations.
Let's start with the vector x. Its length, denoted as ||x||, is given by the square root of the sum of the squared elements of x. Mathematically, this is expressed as ||x|| = √(x_1^2 + x_2^2 + ... + x_n^2), where x_i represents the i-th element of x.
Now, let's consider the vector Ax. This vector is obtained by multiplying the matrix A with the vector x. Therefore, its i-th element is given by the dot product of the i-th row of A and the vector x. Mathematically, this can be written as (Ax)_i = A_i · x, where A_i represents the i-th row of A.
The length of Ax is then calculated using the same formula as before:
||(Ax)|| = √((Ax)_1^2 + (Ax)_2^2 + ... + (Ax)_m^2),
where m represents the number of rows in the matrix A.
Now, let's consider the vector A^T x. This vector is obtained by multiplying A^T (the transpose of A) with the vector x. Therefore, its i-th element is given by the dot product of the i-th row of A^T (which is the i-th column of A) and the vector x. Mathematically, this can be written as (A^T x)_i = A^T_i · x, where A^T_i represents the i-th column of A^T.
The length of A^T x can be calculated using the same formula as before:
||(A^T x)|| = √((A^T x)_1^2 + (A^T x)_2^2 + ... + (A^T x)_n^2),
where n represents the number of columns in the matrix A.
Now, let's use the given fact that AA^T = A^T A.
We can rewrite the dot product (Ax)_i = A_i · x as (Ax)_i = (AA^T)_i · x.
Similarly, we can rewrite the dot product (A^T x)_i = A^T_i · x as (A^T x)_i = (A^T A)_i · x.
Since (AA^T)_i = (A^T A)_i (by the given condition), we can conclude that (Ax)_i = (A^T x)_i for every i.
This means that each element of Ax is equal to the corresponding element of A^T x.
Hence, the two vectors, Ax and A^T x, have the same length.
Therefore, we have shown that the length of Ax equals the length of A^T x when AA^T = A^T A.