In this task, we derived the transformation matrix that would transform 2 vectors to 2 other vectors. The transformation matrix is given by: $$ \begin{bmatrix} -0.5 & 1 \ 1 & 0 \end{bmatrix} $$ The transformation matrix is derived by multiplying the inverse of the matrix formed by the 2 vectors to the matrix formed by the 2 other vectors.
We generated random invertible matrices and random symmetric invertible matrices and reconstructed them from their eigen decomposition.
An invertible matrix can be generated
- by applying random elementery row operations on the identity matrix. The determinant will be non-zero.
And a symmetric matrix can be generated
- by multiplying a matrix with its transpose. if the original matrix is invertible, the symmetric matrix will also be invertible. $$ Det(A\cdot A^T) = Det(A)\cdot Det(A^T) = Det(A)\cdot Det(A) = (Det(A))^2 $$
A matrix can be reconstructed from its eigen decomposition by the following formula: $$ A = X\cdot \lambda \cdot X^{-1} $$
where X is the matrix formed by the eigenvectors of A and
- To decompose the invertible matrix, the
numpyp.linalg.eigfunction was used. - To decompose the symmetric invertible matrix, the
numpy.linalg.eighfunction was used. Becauseeighuses a faster algorithm that takes advantage of the fact that the matrix is symmetric.
We reconstructed a greyscale image from its low rank approximation using SVD. The resulting images with varying K values are shown below:
- Finding algorithms that ensure the invertibility of a matrix
- Finding algorithms that ensure the symmetry of a matrix
- Making image storage efficient using low rank approximation (k(m+n+1) space instead of mn space)