91Ó°ÊÓ

To understand when a matrix is diagonalizable, it is important to start with the concept of eigenvalues. Eigenvalues are scalars associated with a matrix that provide important information about the matrix’s behavior. To find an eigenvalue of a matrix \(A\), we solve the characteristic equation \(\det(A - \lambda I) = 0\).
This involves subtracting \(\lambda\) times the identity matrix \(I\) from \(A\) and computing the determinant. For the matrix given, \(A = \begin{bmatrix} 1 & k \ 0 & 1 \end{bmatrix}\), we computed \(A - \lambda I = \begin{bmatrix} 1 - \lambda & k \ 0 & 1 - \lambda \end{bmatrix}\). The determinant is \((1-\lambda)^2\), leading us to the eigenvalue \(\lambda = 1\).
Eigenvalues tell us about the scaling factor effects of transformations that the matrix represents.

The next step in understanding a matrix's diagonalizability is algebraic multiplicity. This concept refers to the number of times an eigenvalue appears as a root of the characteristic equation.
In this exercise, the equation \((1-\lambda)^2 = 0\) shows that \(\lambda = 1\) is a repeated root, implying an algebraic multiplicity of 2. Accordingly, we can say that this eigenvalue appears "twice".
Algebraic multiplicity is crucial because it gives a preliminary indication of how frequently an eigenvalue influences the matrix. In the context of diagonalization, it also suggests how many times we expect associated eigenvectors may span the eigenspace.

Geometric multiplicity involves how many independent eigenvectors correspond to an eigenvalue.
It is determined by the dimension of the eigenspace corresponding to that eigenvalue. For our exercise matrix, the eigenvector equation \((A - \lambda I)\mathbf{v} = 0\) becomes \(\begin{bmatrix} 0 & k \ 0 & 0 \end{bmatrix}\begin{bmatrix} x \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\). Here, the eigenspace is one-dimensional and spanned by vectors of the form \(\begin{bmatrix} x \ 0 \end{bmatrix}\), where \(x\) is arbitrary. This shows that the geometric multiplicity is 1.
Understanding geometric multiplicity is key for diagonalization: a matrix is diagonalizable if its geometric multiplicity equals its algebraic multiplicity for each eigenvalue.

Eigenvectors are vectors that do not change direction during the matrix transformation, merely getting scaled. They are associated with their corresponding eigenvalues.
For the matrix \(A\), finding eigenvectors involved solving \((A-I)\mathbf{v} = 0\), which reduces due to zero rows giving \(x\) any arbitrary value while \(y = 0\). This indicates a non-unique set of eigenvectors span the eigenspace.
Eigenvectors provide a deeper insight into the matrix, illustrating how a transformation acts along particular directions in space. They are essential for diagonalization because a matrix is diagonalizable if there is a complete set of linearly independent eigenvectors. For our matrix, since only one linearly independent eigenvector was found for an algebraic multiplicity of 2, this signals the matrix is not diagonalizable.