Math 318 Homework 2

(1) Permutation Matrices (6.1 #34) Consider the following permutation matrix:

P =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Let’s examine why P is called a “permutation matrix” and study its eigenvalues and eigenvectors.

(a) What is the effect of multiplying the vector x = (x1,x2,x3,x4) ⊺ by the matrix

P , i.e., what is Px? (b) Find the matrix Q that permutes the entries of a vector x by the permutation

1324. (c) Compute all eigenvalues of the matrix P shown above. (d) For each eigenvalue, find a corresponding eigenvector. You shouldn’t have to

do any tedious computations such as finding nullspaces.

(2) Products and sums of eigenvalues Let A be an n×n matrix with eigenvalues λ1, . . . ,λn (not necessarily all distinct).

(a) (i) (6.2 #20) Suppose A is diagonalizable. Then show that det(A) = λ1λ2⋯λn.

(ii) (6.1 #16) Argue that for any matrix A, det(A) = λ1λ2⋯λn. Hint: Recall that det(A− tI) = (−1)n(t−λ1)(t−λ2)⋯(t−λn).

(b) (6.2 #20, #21) The trace of a square matrix is the sum of its diagonal entries.

For example, if A = [ a b c d

], then trace(A) = a+d.

(i) Argue that trace(PQ) = trace(QP) for any two n×n matrices P and Q. Hint: To prove the general case, write down P = (pij) and Q = (qij) symbolically and compute the trace of PQ and QP symbolically. The trace expression will be a sum of sums. You then need to argue that these sums are the same. You might first check the statement for two symbolic 2 × 2 matrices to understand what is happening.

(ii) Whether you can do (a) or not, use the result to show that if A is diagonalizable, then the trace of A is the sum of the eigenvalues of A.

(iii) In general, the trace of any n×n matrix is the sum of its eigenvalues. (No need to argue this.) Show that the trace of any 3 × 3 matrix is the sum of its eigenvalues. (This requires starting with a symbolic 3 × 3 matrix.) Hint: Here is a hint for general n that you can also use for the n = 3 case. Recall that the characteristic polynomial p(λ) = (−1)n(λ−λ1)⋯(λ−λn) and also p(λ) = det(A−λI). Think about where you might see the sum of the eigenvalues of A if you expanded the product, and where you might see it if you computed det(A−λI) using permutations.

(3) Projections † (6.2 #25) Recall that the column space of a n×n matrix A, denoted Col(A), is

the span of the columns of A. Suppose A is a non-zero n×n matrix such that A2 = A. (a) Show that λ = 1 is an eigenvalue of A with eigenspace equal to Col(A). (b) What is the eigenspace of λ = 1 if A is invertible? Is A diagonalizable in this

case? If yes, write its diagonalization. (c) If A is not invertible, then what are its eigenvalues and their eigenspaces? Is A

diagonalizable in this case? If yes, write its diagonalization. Note: If you can write its diagonalization, it will not be with explicit vectors, but rather a general diagonalization in terms of vectors coming from the various eigenspaces that you have found.

There will be a series of problems throughout this quarter marked with a †. They all relate to projections.

(4) (Simultaneous diagonalization). (6.2 #24, #29) (a) Consider the set of all 4 × 4 matrices that are diagonalized by the same

eigenvector matrix U:

SU = {UΛU −1 ∈ R4×4 ∶ Λ is a diagonal matrix } .

Show that SU is a subspace of R4×4. (Check the properties of a subspace.) (b) What is SI where I is the identity matrix? (c) Suppose the same U diagonalizes A and B, i.e., A = UΛ1U

−1 and B = UΛ2U −1.

Argue that AB = BA. (Recall that in general matrix multiplication is not commutative, i.e., AB ≠ BA.)

(5) * How to find a triangle in a graph. A graph G is a collection of nodes labeled 1, 2, . . . ,n and edges which are pairs of nodes. Shown below is a graph with 5 nodes labeled 1, . . . , 5 and edges {1, 2},{1, 3},{2, 4},{3, 4},{3, 5},{4, 5}. A triangle in a graph G is a triple of nodes i,j,k such that all edges {i,j},{i,k},{j,k} are present in G. For example 3, 4, 5 forms a triangle in the graph below. Two nodes i and j are neighbors in G if {i,j} is an edge in G.

1

2

3

4

5

If the graph G is very large it becomes hard to decide if there is a triangle in it by simply looking at the graph. In this problem we will see that linear algebra can be used to decide if a graph contains a triangle.

(a) The adjacency matrix A of a graph G with n nodes is a n×n matrix defined as follows. The rows and columns of A are indexed by the node labels 1, . . . ,n and the (i,j)-entry of A is 1 if the pair {i,j} is an edge in G and 0 otherwise. (An example can be seen in Problem 2.4A on page 76 in Strang’s book. See also Chapter 3 of the lecture notes.) Write down the adjacency matrix of the graph G shown above.

(b) Let B = A2 where A is the adjacency matrix of G. The entry bij in B is the dot product of two vectors sitting in A. Which vectors are they? In our example, how is b23 formed?

(c) For three nodes i,j,k from G, argue that aikakj is 1 exactly when k is a common neighbor of i and j.

(d) Using the above, what does bij count?

(e) The nodes i,j,k form a triangle if and only if i and j are neighbors and k is a common neighbor of i and j. If i,j,k is a triangle what property must aij and bij have?

(f) Putting all this together can you construct an algorithm that takes as input two indices i,j, and determines whether or not there exists a triangle with the edge between i and j as one of the sides . Justify your algorithm. Think of an algorithm that uses A and B. If you use only A you will likely have a less efficient algorithm. You don’t need to write computer code, just writing out the steps in words is enough.

(g) (optional) If you know about running times of algorithms, do you see how fast this algorithm runs? Is it faster than checking all triples of nodes in G?

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