Find the determinant \(\det A\text{.}\) What does this tell us about the solution space to the homogeneous equation \(A\xvec
= \zerovec\text{?}\)
(b)
Find a basis for \(\nul(A)\text{.}\)
(c)
What is the relationship between the rank of a matrix and the dimension of its null space?
Activity4.2.0.3.
The eigenvalues of a square matrix are defined by the condition that there be a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec=\zerovec\text{.}\)
(a)
If there is a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec = \zerovec\text{,}\) what can we conclude about the invertibility of the matrix \(A-\lambda
I\text{?}\)
\(A-\lambda I\) is invertible.
\(A-\lambda I\) is not invertible.
\(A-\lambda I\) could be either invertible or not invertible, depending on the invertibility of \(A\text{.}\)
(b)
If there is a nonzero solution to the homogeneous equation \((A-\lambda I)\vvec = \zerovec\text{,}\) what can we conclude about the determinant \(\det(A-\lambda I)\text{?}\)
The determinant of \(A-\lambda I\) must be 0.
The determinant of \(A-\lambda I\) must be nonzero.
The determinant of \(A\) must equal the determinant of \(\lambda I\text{.}\)
Find the determinant \(\det(A-\lambda I)\text{.}\)
(b)
What kind of equation do you obtain when we set this determinant to zero to obtain \(\det(A-\lambda I) = 0\text{?}\)
(c)
Use the determinant you found in the previous part to find the eigenvalues \(\lambda\) by solving \(\det(A-\lambda I) = 0\text{.}\) We considered this matrix in the previous section so we should find the same eigenvalues for \(A\) that we found by reasoning geometrically there.
Activity4.2.0.5.
Consider the matrix \(A = \left[\begin{array}{rr}
2 \amp 1 \\
0 \amp 2 \\
\end{array}\right]\) and find its eigenvalues by solving the equation \(\det(A-\lambda I) = 0\text{.}\)
Activity4.2.0.6.
Consider the matrix \(A = \left[\begin{array}{rr}
0 \amp -1 \\
1 \amp 0 \\
\end{array}\right]\) and find its eigenvalues by solving the equation \(\det(A-\lambda I) = 0\text{.}\)
Activity4.2.0.7.
This activity focuses on the eigenvalues of triangular matrices.
(a)
Find the eigenvalues of the triangular matrix \(\left[\begin{array}{rrr}
3 \amp -1 \amp 4 \\
0 \amp -2 \amp 3 \\
0 \amp 0 \amp 1 \\
\end{array}\right]
\text{.}\)
(b)
What is generally true about the eigenvalues of a triangular matrix?
This activity demonstrates a technique that enables us to find the eigenvalues of a square matrix \(A\text{.}\) Since an eigenvalue \(\lambda\) is a scalar for which the equation \((A-\lambda
I)\vvec = \zerovec\) has a nonzero solution, it must be the case that \(A-\lambda I\) is not invertible. Therefore, its determinant is zero. This gives us the equation
Find the eigenvalues and eigenvectors of the diagonal matrix \(A\text{.}\)
(b)
Explain your result by considering the geometric effect of the matrix transformation defined by \(A\text{.}\)
Once we find the eigenvalues of a matrix \(A\text{,}\) describing the eigenspace \(E_\lambda\) amounts to the familiar task of describing the null space \(\nul(A-\lambda I)\text{.}\)
Activity4.2.0.12.
Suppose you have an \(n\times n\) matrix whose characteristic polynomial is \((2-\lambda)^3(-3-\lambda)^{10}(5-\lambda)\text{.}\)
(a)
Identify the eigenvalues, and their multiplicities, of the matrix based on the characteristic polynomial.
(b)
What can you conclude about the dimensions of the eigenspaces?
(c)
What is the dimension of the matrix?
(d)
Do you have enough information to guarantee that there is a basis of \(\real^n\) consisting of eigenvectors?
