Nov 6, 2016 · A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue. Meaning, if you find matrices with distinct …

Jul 26, 2018 · When a matrix is diagonalizable, of course, by definition the diagonal form is similar to the original matrix. Note that similarity holds, more in general, also with the Jordan normal form when the …

Jul 8, 2015 · In short, diagonalizable of a matrix means that the matrix has independent set of eigenvectors that can span the entire space. There are several ways to decide whether matrix has …

But, there are non-diagonalizable matrices that aren't rotations - all non-zero nilpotent matrices. My intuitive view of nilpotent matrices is that they ''gradually collapse all dimensions/gradually lose all the …

I mean, you can say it's similar to a diagonal matrix, it has n independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two linear …

Nov 15, 2017 · There are invertible matrices which are not diagonalizable and there are diagonalizable matrices which are not invertible. There are matrices which are neither diagonalizable nor invertible.

May 15, 2017 · As the other posters comment, there are diagonal matrices which are not multiples of the identity, for example $$ \begin {bmatrix}1&0\\0&2\end {bmatrix} $$ and if all the eigenvalues of a …

Feb 15, 2022 · Diagonalizing Matrix for Normal matrices is a orthogonal matrix. Orthogonal matrices represent rotation. Thus Normal matrices first rotates your vector to their co-ordinate system (given …

The fact that a hermitian matrix is diagonalizable is a corollary of the theorem about normal matrices.

Diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices. He defined a metric (I believe) that was somehow related to the usual metric on $\mathbb {R}^ {n^2}$.