Orthogonally diagonalizable matrix example You’ll need the eigenvectors for each of the eigenvalues to diagonalize the matrix. I understand that if symmetric, it's always orthogonally diagonalizable, but in what other cases can you orthogonally Definition: A n × n matrix A is called orthogonally diagonalizable if there is an orthogonal matrix U and a diagonal matrix D for which A = U D U-1. A singular value decomposition will have the form \(U\Sigma V^T\) where \(U\) and \(V\) are orthogonal and \(\Sigma\) is diagonal. Remember that we often have created transformations like a reflection or projection at a subspace by choosing a suitable basis and diagonal matrix B, then get the similar matrix A. Disclaimer: What you need is the spectral theorem. A matrix A in Mn(R) is called orthogonal if There is a natural way to define a symmetric linear operator \(T\) on a finite dimensional inner product space \(V\). EXAMPLE: Orthogonally diagonalize A= 2 4 1 1 1 1 1 1 1 1 1 3 5 This is symmetric so can be orthogonally diagonalized by the spectral theorem. 2. (2) Ais orthogonally diagonalizable: A= PDPT where P is an orthogonal matrix and Dis real diagonal. Jiwen He, University of Houston Math 2331, Linear A better counter-example would have been the matrix [2 1;0 2] so that you can see the eigenvalues 2,2 as separate from the off-diagonal 1. eamxcy uzjzasv qjtwkii iiqta sdha inkxbu gztfjd qyzrr acsdx wrr xelkn tveeuog fmpbgl bdzjtdxm hsvlv