Question

Find Singular Value Decomposition (SVD) for matrix A =
{{3,1},{2,4}}

Answer 1

To find the Singular Value Decomposition (SVD) of a matrix, calculate the eigenvalues, find the eigenvectors, normalize the columns, and construct the diagonal matrix.

Step-by-step explanation:

The Singular Value Decomposition (SVD) of a matrix is a factorization technique that breaks down a matrix into three separate matrices - U, Σ, and VT. To find the SVD for matrix A = {{3, 1}, {2, 4}}, follow these steps:

1. Calculate the eigenvalues of the product of A and AT. This will give you the diagonal elements of Σ.
2. Find the eigenvectors corresponding to the eigenvalues obtained in the previous step. These eigenvectors will become the columns of matrix U.
3. Normalize the columns of matrix U.
4. Calculate the matrix V by finding the eigenvectors of AT A. Normalize the columns of V.
5. Construct the diagonal matrix Σ using the eigenvalues obtained in step 1.

Therefore, the SVD for matrix A = {{3, 1}, {2, 4}} is U = {{-0.376, -0.927}, {-0.927, 0.376}}, Σ = {{5.237, 0}, {0, 1.372}}, and VT = {{-0.553, -0.833}, {-0.833, 0.553}}.