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 …

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 …

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 …

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 …

Nov 15, 2017 · From my understanding, invertible means non-singular and any of eigenvalue must not be 0. Exactly. In fact, a matrix is singular if and only if $0$ is its eigenvalue. …

4 I'll try an answer in a different (equivalent) direction: what happens when the matrix is not diagonalizable? First of all, this must mean that some of the matrix's eigenvalues occur more …

Regarding the second part of (b), since it can't be true as asked, we could modify it to a more interesting question: If two diagonalizable matrices commute, are they simultaneously …

Feb 15, 2022 · Since a diagonalizable matrix represents a scaling operation under certain basis, so I wonder what additional geometrical meanings a normal matrix processes to be …

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 …

A matrix is diagonalizable if and only if the algebraic multiplicity equals the geometric multiplicity of each eigenvalues. By your computations, the eigenspace of $\lambda=1$ has dimension $1$; …