diagonalize matrix given eigenvalues matlab

Diagonalizing a Matrix Given Eigenvalues in Matlab

When it comes to diagonalizing a matrix given its eigenvalues, Matlab provides the `eig` function, which is capable of orthogonally diagonalizing symmetric matrices. However, when using this function on dense n-by-n matrices, Matlab allocates new storage each time you call it, even if you provide pre-allocated storage.

In contrast, Julia's `eigen!` function calls LAPACK's `geevx!`, which includes boilerplate code that allocates seven matrices similar to the original matrix. This can result in different allocation patterns and timings between Matlab and Julia.

To illustrate this, let's consider a 288-by-288 symmetric matrix A. When we use Matlab's `eig` function on this matrix, it returns the eigenvectors V and eigenvalues D as symbolic matrices. The columns of V represent the right eigenvectors of A, while D is a diagonal matrix with elements approximating the eigenvalues.

In Julia, we can use the `LinearAlgebra` package to diagonalize a matrix given its eigenvalues. For example, if we define a random complex-valued 288-by-288 matrix H and then compute its eigenvectors and eigenvalues using the `eigen` function, we get a diagonal matrix D with elements approximating the eigenvalues and a matrix V whose columns are the corresponding right eigenvectors.

Interestingly, when comparing the timings of Matlab and Julia for this operation, we find that Matlab is about 1 ms faster. However, it's possible that both languages can run the same BLAS (Basic Linear Algebra Subprograms) library and still produce different timings due to differences in their Just-In-Time (JIT) compilers and allocation strategies.

In conclusion, Matlab's `eig` function provides a convenient way to diagonalize symmetric matrices given their eigenvalues. However, when it comes to efficiency, the choice of language and the specific implementation details can have a significant impact on performance.

Diagonalize matrix given eigenvalues matlab