All square matrices are not necessarily diagonalizable.
Consider the example
\(\begin{bmatrix}
0 & 1 \\
0 & 0 \\
\end{bmatrix}\)
Its characteristic polynomial gives use $\lambda ^ 2 = 0$, and thus the only eigenvalue is 0.
Since the polynomial has two identical roots 0, we say that the algebraic multiplicity of the root 0, is 2.
Naturally it follows that the sum of all the algebraic multiplicities of each root must add up to $n$,
since the characteristic polynomial must be of degree $n$, the dimension of the matrix.
The geometric multiplicity is the number of dimensions of $ker(M - \lambda I)$, which is the same as the number of linearly independent eigenvectors with eigenvalue $\lambda$
The norm or the “length” of a vector $v$ is defined as $\sqrt{\langle v, v \rangle}$, the square root of the inner product of the vector with itself. We have already seen the definitions and relevant properties of the inner product in previous lectures.