Question

For each matrix \(A\), find the eigenvalues and eigenvectors. Give the geometric and algebraic multiplicity of each eigenvalue. Does \(A\) have a full set of eigenvectors?\(A=\left[\begin{array}{llll}2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 3\end{array}\right]\)

Short answer

In summary, we have found the eigenvalues and eigenvectors for the given matrix. The eigenvalues are \(\lambda_1 = 2\) with geometric multiplicity 2 and algebraic multiplicity 3, and \(\lambda_2 = 3\) with geometric multiplicity 1 and algebraic multiplicity 1. Since the sum of the dimensions of the eigenspaces is not equal to the size of the matrix, the matrix does not have a full set of eigenvectors.

Step-by-step solution

Calculate the eigenvalues

Determine the eigenvalues by solving the equation for the characteristic polynomial, which is given by the determinant of \((A-\lambda I)\). $A-\lambda I = \left[\begin{array}{llll}2-\lambda & 0 & 0 & 0 \\\ 0 & 2-\lambda & 0 & 0 \\\ 0 & 0 & 2-\lambda & 0 \\\ 0 & 0 & 1& 3-\lambda\end{array}\right]$ Now, we will find the determinant: \(\det(A-\lambda I) = (2-\lambda)((2-\lambda)^3)(3-\lambda)\) Setting this equation equal to zero, we get the eigenvalues: \(\lambda_1 = 2, \lambda_2 = 3\)

Find the eigenvectors

We will now substitute each eigenvalue back into the equation \((A-\lambda I)v=0\) to find the eigenvectors. For \(\lambda_1 = 2:\) $(A-2I)v = \left[\begin{array}{llll}0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 1\end{array}\right]v = 0$ \(v = \left[\begin{array}{c}a_1 \\\ 0a_1\\\ a_2\\\ 0\end{array}\right]\) where \(a_1\) and \(a_2\) are any constants For \(\lambda_2 = 3:\) \((A-3I)v = \left[\begin{array}{llll}-1 & 0 & 0 & 0 \\\ 0 & -1 & 0 & 0 \\\ 0& 0 & -1 & 0 \\\ 0 & 0 & 1 & 0\end{array}\right]v = 0\) \(v = \left[\begin{array}{c}0 \\\ 0\\\ 0\\\ a_3\end{array}\right]\) where \(a_3\) is any constant

Find the geometric multiplicities

The geometric multiplicity of each eigenvalue is the dimension of the eigenspace (null space) of \((A-\lambda I)\). For \(\lambda_1 = 2,\) the eigenspace is spanned by two linearly independent vectors, the geometric multiplicity is 2. For \(\lambda_2 = 3,\) the eigenspace is spanned by only one linearly independent vector, the geometric multiplicity is 1.

Find the algebraic multiplicities

The algebraic multiplicity of each eigenvalue is the largest number of times it can appear as a root in the characteristic polynomial. For \(\lambda_1 = 2,\) the algebraic multiplicity is 3. For \(\lambda_2 = 3,\) the algebraic multiplicity is 1.

Determine if A has a full set of eigenvectors

A matrix \(A\) has a full set of eigenvectors if the sum of the dimensions of the eigenspaces for each eigenvalue is equal to the size of the matrix. In our case, the sum of the dimensions of the eigenspaces is \(2 + 1 = 3\). Since the size of the matrix is 4, \(A\) does not have a full set of eigenvectors.

Key concepts

Characteristic Polynomial

The characteristic polynomial is a fundamental concept when studying eigenvalues of a matrix. To find the characteristic polynomial, one must subtract a scalar multiple of the identity matrix \( \lambda I \) from the matrix \( A \) and then compute the determinant of the resulting matrix, \( A - \lambda I \) . For the matrix in the given exercise, the characteristic polynomial is obtained by calculating the determinant of the subtracted matrix, leading to \( \det(A-\lambda I) = (2-\lambda)^3(3-\lambda) \) .

By setting this polynomial equal to zero, we can find the eigenvalues of the matrix. The roots of the polynomial - the values for \( \lambda \) that make the equation true - are precisely the eigenvalues. This process is crucial because the roots of the characteristic polynomial dictate the behavior of the matrix and its power iterations, influencing the vectors that remain invariant under its application.

Geometric Multiplicity

Geometric multiplicity pertains to the number of linearly independent eigenvectors associated with an eigenvalue. It is determined by the dimension of the eigenspace, which is the null space of \( A - \lambda I \) . In simpler terms, it's the number of 'directions' in which you can move without changing the nature of the transformation induced by the matrix.

For the exercise at hand, we see that the eigenspace corresponding to \( \lambda_1 = 2 \) is spanned by two vectors - thus yielding a geometric multiplicity of '2'. On the other hand, for \( \lambda_2 = 3 \) there's only a single vector which maintains its direction after the transformation, implying a geometric multiplicity of '1'. These multiplicities give us insight into the underlying structure and symmetry of the matrix transformation.

Algebraic Multiplicity

Algebraic multiplicity refers to the number of times an eigenvalue appears as a solution, or 'root', of the characteristic polynomial. It's a measure of the eigenvalue's weight or redundancy within the polynomial. The algebraic multiplicity may differ from geometric multiplicity, revealing important aspects of the matrix's diagonalizability.

In our matrix \( A \) , \( \lambda_1 = 2 \) has an algebraic multiplicity of '3' since it is a triple root of the characteristic polynomial. Conversely, \( \lambda_2 = 3 \) appears just once in the polynomial and hence has an algebraic multiplicity of '1'. These findings are important when considering the potential for a matrix to be similar to a diagonal matrix, which has its eigenvalues laid out on the main diagonal.

Eigenspaces

Eigenspaces are essentially a visual and conceptual representation of all the eigenvectors that relate to a specific eigenvalue. They form vector spaces, which in simple terms is like plotting all possible 'directions' (eigenvectors) that remain unchanged when multiplied by the matrix. Each eigenspace is tied to one eigenvalue and is defined as the null space of \( A - \lambda I \) .

In the context of the given exercise, we have two eigenspaces: one corresponding to \( \lambda_1 \) with a basis composed of two vectors, and one for \( \lambda_2 \) with only one vector in its basis. The dimensions of these spaces are indicative of the matrix's ability to be diagonalized, and the understanding of these spaces can dramatically simplify matrix-related calculations.

Matrix Determinant

The matrix determinant is a scalar attribute associated with matrices that provides a lot of information about the linear transformation the matrix represents. It helps in computing volumes, solving systems of linear equations, and understanding whether a matrix is invertible or not.

In the solution provided, while finding the characteristic polynomial, we calculated the determinant of \( A - \lambda I \) . The determinant of our original matrix \( A \) isn't directly computed, but knowing that eigenvalues are various values of \( \lambda \) that make \( \det(A-\lambda I) = 0 \) is crucial. If the determinant of \( A \) were zero, the matrix would not be invertible, which offers valuable insight, especially in solving systems of linear equations where the matrix \( A \) is the coefficient matrix.