Verify eigenvectors and use a diagonal form to compute a matrix power. Use Cayley–Hamilton to reduce powers and construct an inverse. Separate the singular parameter case before solving a system.
Notes and key methods
An eigenvector \(\displaystyle v\ne 0\) satisfies Av=λv. If a square matrix has a basis of eigenvectors, form Q from those vectors as columns and D from the corresponding eigenvalues in the same order: A=QDQ⁻¹.
Then \(\displaystyle A^{n}=QD^{n}Q^{−1}\), and the trace and determinant are respectively the sum and product of eigenvalues. Distinct real eigenvalues in these examples guarantee a real eigenvector basis.
The characteristic polynomial is det(λI−A). Cayley–Hamilton states that substituting A into this polynomial gives the zero matrix. A higher power can therefore be expressed as a combination of lower powers.
When A is invertible, multiply the characteristic equation by \(\displaystyle A^{−1}\) to obtain an inverse formula. Identity matrices are essential: a constant term in a matrix polynomial means that constant times I.
When a parameter occurs in a linear system, first factor the determinant. Solve the nonsingular cases normally, then put each exceptional parameter back into the augmented equations.
Consistency in a singular case requires that the constants obey the same linear relation as the corresponding coefficient rows. With consistent dependent equations, exhibit a free variable rather than choosing it silently.
Check the conditions
An eigenvector is nonzero.
Match the order of Q columns to the order of diagonal entries in D.
\(\displaystyle A^{−1}\) is available only when det \(\displaystyle A\ne 0\).
Use a consistent det(λI−A) sign convention.
Exceptional parameter values must be analysed separately.
A row dependency must also hold for the corresponding right-hand sides.
\(A^2=pA-qI\); apply this relation twice to obtain \(A^4=\left[\begin{matrix}1538 & 1131\\1131 & 2669\end{matrix}\right]\).
Answer: \(1538\).
Example 3 · Parameter-dependent consistency
Let \(A(t)=\left[\begin{matrix}1 & 1 & 1\\1 & t - 5 & 2\\2 & 2 & t - 4\end{matrix}\right]\). In the singular case consider right-hand side \((8,15,c)^T\). For the separate nonsingular calculation set \(t=8\) and use \(b=\left[\begin{matrix}22\\45\\62\end{matrix}\right]\).
Find the singular value of t.
Find the value of c that makes the singular system consistent.
Find x in the specified nonsingular system.
Find the determinant at \(\displaystyle t=14\).
Show the reasoning
(a)
\(\det A(t)=\left(t - 6\right)^{2}=0\)
Answer: \(6\).
(b)
\(c=2(8)=16\) because the third coefficient row is twice the first.
Answer: \(16\).
(c)
\(v=\left[\begin{matrix}6\\7\\9\end{matrix}\right]\) by elimination.
Answer: \(6\).
(d)
\((14-6)^2=64\)
Answer: \(64\).
Common mistakes
Eigenvectors are columns, not rows, in A=QDQ⁻¹.
Do not raise each entry of A separately to compute \(\displaystyle A^{n}\).
A scalar constant in a matrix identity is multiplied by I.
An entrywise reciprocal is not a matrix inverse.
Do not cancel a parameter factor and lose a singular case.
A singular system can have either no solution or infinitely many.