University Linear Algebra — Eigenvalues and Diagonalization Worksheet

All 13 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Eigenvalues and Eigenvectors from the Definition

   Let $$A=\left[\begin{array}{cccc}3& -4& -4& -4\\ 1& 4& 2& 2\\ 0& -3& -1& -3\\ -1& -2& -2& 0\end{array}\right].$$

   1. Working from the definition only (no characteristic polynomial), decide which of the following are eigenvectors of $A$, and give the eigenvalue of each one that is: $$\overrightarrow{u}=(1,1,0,-1),\overrightarrow{v}=(0,-1,-1,2),\overrightarrow{w}=(1,0,0,0),\overrightarrow{0}=(0,0,0,0).$$
   2. Decide whether $\lambda =1$ is an eigenvalue of $A$.
2. Q2Eigenvalues and Eigenvectors from the Definition

   Let $V$ be the vector space of functions $f:ℝ\to ℝ$ that have derivatives of every order, and let $D:V\to V$ be differentiation, $D(f)=f^{\prime }$. An eigenvector of a linear operator $T$ is a nonzero $f$ with $T(f)=\lambda f$ for some real $\lambda$. For each of $$f_1(x)=e^{-4x},f_2(x)=x{e}^x,f_3(x)=5,f_4(x)=\cos 2x,$$ decide whether it is an eigenvector of $D$, and whether it is an eigenvector of $D^2=D\circ D$. Give the eigenvalue whenever it is one.

3. Q3The Characteristic Polynomial

   The characteristic polynomial of a square matrix $A$ is $p(\lambda )=det(A-\lambda I)$. Let $$A=\left[\begin{array}{ccc}4& 0& 1\\ 2& 3& 2\\ 1& 0& 4\end{array}\right].$$

   1. Find $p(\lambda )$ in factored form.
   2. List the eigenvalues of $A$ with their algebraic multiplicities.
   3. Check that the eigenvalues, each counted as often as its multiplicity, add up to $tr(A)$ and multiply to $det(A)$.
4. Q4The Characteristic Polynomial

   Let $$A=\left[\begin{array}{cccc}1& 2& 5& -1\\ 3& 2& 0& 4\\ 0& 0& 2& 1\\ 0& 0& 1& 2\end{array}\right].$$ Find the characteristic polynomial $det(A-\lambda I)$ in factored form and the eigenvalues of $A$. Check your eigenvalues against $tr(A)$.

5. Q5Eigenspaces and Their Dimensions

   Let $$A=\left[\begin{array}{cccc}2& 1& 0& 1\\ 3& 2& 0& 3\\ -3& 1& 2& -2\\ -3& -1& 0& -2\end{array}\right].$$ Given that $\lambda =2$ is an eigenvalue of $A$:

   1. find a basis for the eigenspace $E_2$ and state $\dim E_2$;
   2. write $E_2$ in set-builder notation.
6. Q6Eigenspaces and Their Dimensions

   Let $T:M_{2\times 2}\to M_{2\times 2}$ be the linear map $T(X)=X^T$.

   1. Show that if $T(X)=\lambda X$ for a nonzero $X$, then $\lambda =1$ or $\lambda =-1$. (Apply $T$ twice.)
   2. Describe the eigenspaces $E_1$ and $E_{-1}$ in set-builder notation, and give a basis and the dimension of each.
7. Q7Diagonalizing a Matrix

   Let $$A=\left[\begin{array}{ccc}2& 0& -1\\ 3& -1& -3\\ 0& 0& 1\end{array}\right].$$ Find an invertible matrix $P$ and a diagonal matrix $D$ with $A=PD{P}^{-1}$. Check your answer by verifying $AP=PD$.

8. Q8Diagonalizing a Matrix

   Let $T:P_2\to P_2$ be the linear map $T(p(x))=p(2x+1)$, where $P_2$ is the space of polynomials of degree at most $2$.

   1. Find the matrix $[T{]}_{ℬ}$ of $T$ relative to the basis $ℬ=\{1,x,x^2\}$, and its eigenvalues.
   2. Find a basis $𝒞$ of $P_2$ consisting of eigenvectors of $T$, and write down $[T{]}_{𝒞}$.
9. Q9Deciding Whether a Matrix Is Diagonalizable

   Each matrix below is given with its characteristic polynomial $det(M-\lambda I)$. Decide whether it is diagonalizable, and justify.

   1. $A=\left[\begin{array}{cccc}4& 5& 0& -5\\ -3& -4& -2& 5\\ 0& 4& 3& -4\\ -3& -3& -2& 4\end{array}\right]$, $(\lambda -1)(\lambda +1)(\lambda -3)(\lambda -4)$.
   2. $B=\left[\begin{array}{ccc}2& 0& 1\\ -3& 5& 1\\ 0& 0& 2\end{array}\right]$, $-(\lambda -2{)}^2(\lambda -5)$.
   3. $C=\left[\begin{array}{ccc}2& 0& 0\\ -3& 5& 3\\ 0& 0& 2\end{array}\right]$, $-(\lambda -2{)}^2(\lambda -5)$.
10. Q10Deciding Whether a Matrix Is Diagonalizable

    For which real numbers $k$ is $$A=\left[\begin{array}{cccc}2& k& 1& 0\\ 0& 2& 0& 0\\ 0& 0& 3& 0\\ 0& 0& 0& 2\end{array}\right]$$ diagonalizable? Justify.

11. Q11Powers of a Diagonalizable Matrix

    Let $A=\left[\begin{array}{cc}0& 2\\ 1& 1\end{array}\right]$.

    1. Diagonalize $A$.
    2. Find a formula for $A^n$, $n\ge 1$, and use it to write down $A^{10}$.
12. Q12Complex Eigenvalues of a Real Matrix

    Let $A=\left[\begin{array}{cc}1& -2\\ 1& 3\end{array}\right]$, and work with real scalars.

    1. Find the roots of the characteristic polynomial $det(A-\lambda I)$.
    2. Does $A$ have a real eigenvector? Is $A$ diagonalizable over $ℝ$?
    3. Is there a line through the origin in ${ℝ}^2$ that multiplication by $A$ maps into itself? Explain.
13. Q13Synthesis — drawing on several topics in this unit

    Let $$A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& -1\\ 1& 0& 0\end{array}\right].$$

    1. Find the characteristic polynomial $det(A-\lambda I)$, and the eigenvalues with their algebraic multiplicities.
    2. Find a basis for each eigenspace.
    3. Decide whether $A$ is diagonalizable; if it is, give $P$ and $D$ with $A=PD{P}^{-1}$.
    4. Use (c) to find $A^{50}$.

The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Linear Algebra bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Eigenvalues and Diagonalization notes cover  About the Eigenvalues and Diagonalization unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Eigenvectors from the definition

A non-zero vector v is an eigenvector of A when Av is a multiple of v; the multiple is the eigenvalue. Q1 asks you to decide for four vectors without computing any polynomial. That means multiplying: compute Av and compare it with v, entry by entry.

Q1(b) asks whether a given number is an eigenvalue. That is the question of whether A − λI has a non-trivial null space — equivalently, whether it fails to be invertible. Row reduce it and look for a free variable, or compute its determinant.

Q2 asks the same question of differentiation, acting on functions. A function is an eigenvector of D when its derivative is a constant multiple of itself — differentiate each one and see whether the result is a multiple of the original. For D², differentiate twice. A function can be an eigenvector of D² without being one of D, so decide the two questions separately for each function.

The characteristic polynomial

The eigenvalues are the roots of det(A − λI). Q3 asks for it in factored form, and the choice of expansion decides how much work that is.

Q3(c) is a check you should use every time: counted with multiplicity, the eigenvalues add to the trace and multiply to the determinant.

Q4 is a 4 × 4 matrix with a block of zeros in its lower left corner. A block triangular matrix has determinant equal to the product of the determinants of its diagonal blocks, and that applies to A − λI too — so the characteristic polynomial is the product of two quadratics.

Eigenspaces

The eigenspace of λ is the null space of A − λI: every eigenvector for λ, together with zero. Q5 gives the eigenvalue and asks for a basis and the dimension. Form A − 2I, reduce, and read off one vector per free variable, exactly as in the rank set. Part (b) writes the same space as a set defined by conditions — the equations of the reduced system.

Q6 is an operator on 2 × 2 matrices: the transpose. Part (a) uses the hint — applying the transpose twice gives the original matrix, so if T(X) = λX then applying T again gives λ²X, which must equal X. Part (b) asks which matrices satisfy X^T = X and which satisfy X^T = −X; describe each set by conditions on the entries, then find a basis by the method of the Basis set.

Diagonalizing

A matrix is diagonalizable when it has enough independent eigenvectors to form a basis. Then A = PDP⁻¹, where the columns of P are those eigenvectors and D has the matching eigenvalues on its diagonal. It is the change-of-basis formula from the previous set, with the eigenvector basis chosen so that the new matrix is diagonal.

Q7 is the procedure on a 3 × 3 matrix. Q8 applies it to a map on quadratics: build its matrix relative to the standard basis as in the previous set, find the eigenvalues — the matrix is triangular, so they are on its diagonal — then find eigenvectors and translate each back into a polynomial. Relative to a basis of those polynomials, the matrix of the map is diagonal.

When is a matrix diagonalizable?

Q9 gives three matrices with their characteristic polynomials and asks for a verdict with a justification. The polynomial alone decides some cases; others need an eigenspace dimension.

Q10 puts a parameter into the same test. The matrix is triangular, so its eigenvalues can be read from the diagonal, and one of them repeats. The question is how the entry k changes the rank of A − λI for that repeated eigenvalue — compute the rank as a function of k, and split into cases.

Powers of a diagonalizable matrix

If A = PDP⁻¹, then in any power of A the inner P⁻¹P pairs cancel, and a power of a diagonal matrix is just the powers of its diagonal entries.

Q11 asks for the diagonalization of a 2 × 2 matrix, then a formula for its n-th power valid for every n, then a specific power from the formula. Keep n as a letter until the last step — the question wants a formula, not a number. A quick check: your formula must give A itself when n = 1.

Complex eigenvalues of a real matrix

Q12 is a real 2 × 2 matrix whose characteristic polynomial is a quadratic. Part (a) is the quadratic formula; look at the discriminant first. Parts (b) and (c) ask what happens when the roots are not real numbers.

The synthesis question

Q13 runs the whole set on one 3 × 3 matrix. Part (a) is Q3 — expand along a line with zeros, keep factors, and check the trace. Part (b) is Q5 for each eigenvalue. Part (c) is Q9's test: compare, for the repeated eigenvalue, the multiplicity with the dimension you just found, then assemble P and D if the test passes. Part (d) is Q11 — use the diagonal form, and notice what happens to powers of the eigenvalues in D before multiplying anything out.

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Getting the most out of it

Check every eigenvector by multiplying

Whatever method produced it, an eigenvector is correct exactly when Av = λv. The check is one matrix-vector product, and it catches both a wrong eigenvalue and a wrong null-space vector.

Use trace and determinant as a checksum

After factoring a characteristic polynomial, add the roots and compare with the trace. It takes seconds and catches most sign errors in the determinant.

Keep P and D in matching order

Write the eigenvalues in D in the same order as their eigenvectors in P. Any order works, but the two must match — a mismatch gives a matrix that is not A.

Count before you assemble

Before building P, add up the eigenspace dimensions. If they fall short of n, the matrix is not diagonalizable, and the question is answered.

Common questions