Lesson test: Diagonalisation and powers

1.[2p]
In $A = P D P^{- 1}$ , what are the columns of $P$ and the diagonal entries of $D$ ?

The eigenvectors of $A$ , and its eigenvalues in the same order

Correct

The rows of $A$ , and its pivots

The answer is: The eigenvectors of $A$ , and its eigenvalues in the same order

The columns of $A$ , and its determinant

The answer is: The eigenvectors of $A$ , and its eigenvalues in the same order

Any basis, and any scalars

The answer is: The eigenvectors of $A$ , and its eigenvalues in the same order

2.[2p]
Why is $A^{k} = P D^{k} P^{- 1}$ ?

The inner factors $P^{- 1} P$ cancel each time the product is written out

Correct

Because $D$ commutes with every matrix

The answer is: The inner factors $P^{- 1} P$ cancel each time the product is written out

Because $P$ is orthogonal

The answer is: The inner factors $P^{- 1} P$ cancel each time the product is written out

It holds only for $k = 2$

The answer is: The inner factors $P^{- 1} P$ cancel each time the product is written out

3.[3p]
$C = [(2, 1), (1, 2)]$ has $C^{k}$ with diagonal entries $(3^{k} + 1) / 2$ . What is the diagonal entry of $C^{4}$ ?

CorrectNot quite: 41

4.[3p]
For the Markov matrix $[(0.9, 0.1), (0.2, 0.8)]$ , what is the long-run share of the first state, as a decimal to three places?

CorrectNot quite: 0.667

5.[2p]
For that same chain, what is the second eigenvalue, which sets the speed of convergence?

CorrectNot quite: 0.7

6.[2p]
Every Markov matrix, whose columns sum to one, has an eigenvalue of exactly one.

True

Correct

False

The answer is: True

7.[3p]
A matrix with $n$ distinct eigenvalues is always diagonalisable because

eigenvectors belonging to distinct eigenvalues are independent, so they form a basis

Correct

distinct eigenvalues force the matrix to be symmetric

The answer is: eigenvectors belonging to distinct eigenvalues are independent, so they form a basis

the determinant is then non-zero

The answer is: eigenvectors belonging to distinct eigenvalues are independent, so they form a basis

the characteristic polynomial then factors over the reals

The answer is: eigenvectors belonging to distinct eigenvalues are independent, so they form a basis

8.[2p]
The Fibonacci matrix $[(1, 1), (1, 0)]$ has largest eigenvalue $φ$ . What is it, to three decimal places?

CorrectNot quite: 1.618

9.[3p]
Which matrices fail to be diagonalisable over the real numbers?

Select all that apply

The shear $[(1, 0), (1, 1)]$

Correct

A quarter turn rotation

Correct

The identity

The answer is: The shear $[(1, 0), (1, 1)]$ , A quarter turn rotation

Any matrix with two distinct real eigenvalues

The answer is: The shear $[(1, 0), (1, 1)]$ , A quarter turn rotation

1.[2p] In A=PDP−1, what are the columns of P and the diagonal entries of D?

2.[2p] Why is Ak=PDkP−1?

3.[3p] C=[(2,1),(1,2)] has Ck with diagonal entries (3k+1)/2. What is the diagonal entry of C4?

4.[3p] For the Markov matrix [(0.9,0.1),(0.2,0.8)], what is the long-run share of the first state, as a decimal to three places?

5.[2p] For that same chain, what is the second eigenvalue, which sets the speed of convergence?

6.[2p] Every Markov matrix, whose columns sum to one, has an eigenvalue of exactly one.

7.[3p] A matrix with n distinct eigenvalues is always diagonalisable because

8.[2p] The Fibonacci matrix [(1,1),(1,0)] has largest eigenvalue φ. What is it, to three decimal places?

9.[3p] Which matrices fail to be diagonalisable over the real numbers?

1.[2p]
In $A = P D P^{- 1}$ , what are the columns of $P$ and the diagonal entries of $D$ ?

2.[2p]
Why is $A^{k} = P D^{k} P^{- 1}$ ?

3.[3p]
$C = [(2, 1), (1, 2)]$ has $C^{k}$ with diagonal entries $(3^{k} + 1) / 2$ . What is the diagonal entry of $C^{4}$ ?

4.[3p]
For the Markov matrix $[(0.9, 0.1), (0.2, 0.8)]$ , what is the long-run share of the first state, as a decimal to three places?

5.[2p]
For that same chain, what is the second eigenvalue, which sets the speed of convergence?

6.[2p]
Every Markov matrix, whose columns sum to one, has an eigenvalue of exactly one.

7.[3p]
A matrix with $n$ distinct eigenvalues is always diagonalisable because

8.[2p]
The Fibonacci matrix $[(1, 1), (1, 0)]$ has largest eigenvalue $φ$ . What is it, to three decimal places?

9.[3p]
Which matrices fail to be diagonalisable over the real numbers?