Chapter 14

Nonnormal Matrices

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Section 14.1Chapter Problems[22/100%]
Problem 14.1.1

Let AA be a nonnormal n×nn \times n matrix over C\mathbb {C}.

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(i)

Show that A+A∗A+A^{*} is a normal matrix.

(ii)

Show that AA∗AA^{*} is a normal matrix.

Problem 14.1.2

Let AA be a nonnormal matrix. Show that [A,A∗][A,A^{*}] is a normal matrix. Show that [A,A∗]+[A,A^{*}]_{+} is a normal matrix.

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Problem 14.1.3

Let α,β∈C\alpha , \beta \in \mathbb {C}. What are the conditions on α\alpha, β\beta such that the matrix

M(α,β)=(αβ01)⇒M∗(α,β)=(αˉ0βˉ1) M(\alpha ,\beta ) = \begin{pmatrix} \alpha & \beta \\ 0 & 1 \end{pmatrix} \Rightarrow M^{*}(\alpha ,\beta ) = \begin{pmatrix} \bar\alpha & 0 \\ \bar\beta & 1 \end{pmatrix}

is normal?

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Problem 14.1.4

Let θ∈[0,2π)\theta \in [0,2\pi ). What are the conditions on θ\theta such that the matrix A(θ)A(\theta )

A(θ)=(0sin⁡(θ)cos⁡(θ)0)⇒A∗(θ)=(0cos⁡(θ)sin⁡(θ)0) A(\theta ) = \begin{pmatrix} 0 & \sin (\theta ) \\ \cos (\theta ) & 0 \end{pmatrix} \Rightarrow A^{*}(\theta ) = \begin{pmatrix} 0 & \cos (\theta ) \\ \sin (\theta ) & 0 \end{pmatrix}

is normal?

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Problem 14.1.5

Let θ∈R\theta \in \mathbb {R}. What are the conditions on θ\theta such that the matrix A(θ)A(\theta )

A(θ)=(0sinh⁡(θ)cosh⁡(θ)0)⇒A∗(θ)=(0cosh⁡(θ)sinh⁡(θ)0) A(\theta ) = \begin{pmatrix} 0 & \sinh (\theta ) \\ \cosh (\theta ) & 0 \end{pmatrix} \Rightarrow A^{*}(\theta ) = \begin{pmatrix} 0 & \cosh (\theta ) \\ \sinh (\theta ) & 0 \end{pmatrix}

is normal?

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Problem 14.1.6

Let z∈Cz \in \mathbb {C} and z≠0z \neq 0. Show that the 2×22 \times 2 matrix

A=(1z0−1) A = \begin{pmatrix} 1 & z \\ 0 & -1 \end{pmatrix}

is nonnormal. Give the eigenvalues.

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Problem 14.1.7
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(i)

Can we conclude that an invertible matrix AA is normal?

(ii)

Let BB be an n×nn \times n matrix over C\mathbb {C} with B2=InB^{2} = I_{n}. Can we conclude that BB is normal?

Problem 14.1.8

Show that not all nonnormal matrices are non-diagonalizable.

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Problem 14.1.9

Find all 2×22 \times 2 matrices over C\mathbb {C} which are nonnormal but diagonalizable.

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Problem 14.1.10

Let AA be a normal n×nn \times n matrix and let SS be an invertible nonnormal matrix. Is SAS−1SAS^{-1} a normal matrix? Study the case

A=12(111−1),S=(1−101). A = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad S = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}.

Of course the eigenvalues of AA and SAS−1SAS^{-1} are the same.

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Problem 14.1.11

Consider the nonnormal 2×22 \times 2 matrix

A=(0100). A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.

Find sinh⁡(A)\sinh (A) and tanh⁡−1(A)\tanh^{-1}(A).

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Problem 14.1.12
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(i)

Consider the nonnormal 2×22 \times 2 matrix

A=(0i00)⇒A∗=(00−i0). A = \begin{pmatrix} 0 & i \\ 0 & 0 \end{pmatrix} \Rightarrow A^{*} = \begin{pmatrix} 0 & 0 \\ -i & 0 \end{pmatrix}.

Find a unitary matrix UU such that UAU∗=A∗UAU^{*} = A^{*}.

(ii)

Consider the nonnormal 2×22 \times 2 matrix

K=(−ii0i). K = \begin{pmatrix} -i & i \\ 0 & i \end{pmatrix}.

Find a unitary matrix VV such VKV∗=K∗VKV^{*} = K^{*}. Find the eigenvalues and normalized eigenvectors of KK and K∗K^{*}. Discuss.

Problem 14.1.13

Let AA, BB be normal matrices. Can we conclude that ABAB is a normal matrix?

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Problem 14.1.14

Consider the nonnormal 3×33 \times 3 matrix

B=(011011000). B = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix}.

Can we find a 3×33 \times 3 matrix YY such that Y2=BY^{2} = B, i.e. YY would be the square root of BB.

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Problem 14.1.15

Let AA be a normal n×nn \times n matrix. Is the matrix A−iInA-iI_{n} normal?

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Problem 14.1.16

A measure of nonnormality is given by

m(A):=∥A∗A−AA∗∥ m(A) := \left\| A^{*}A-AA^{*}\right\|

where ∥⋅∥\left\| \cdot \right\| denotes some matrix norm. Let S3S_{3} and S1S_{1} be real-valued and symmetric n×nn \times n matrices (for example, spin matrices). Calculate m(S3+exp(iϕ)S1)m(S_{3}+\mathrm{exp}\left(i\phi \right)S_{1}).

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Problem 14.1.17

Consider the nonnormal matrix

A=(iπ10−iπ). A = \begin{pmatrix} i\pi & 1 \\ 0 & -i\pi \end{pmatrix}.

Is eA+A∗=eAeA∗e^{A+A^{*}} = e^{A}e^{A^{*}}?

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Problem 14.1.18

Prove or disprove the following statements.

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(i)

If the n×nn \times n matrix AA is nonnormal, then there exists no matrix BB such that B2=AB^{2} = A.

(ii)

Let AA, BB be nonzero n×nn \times n matrices with B2=AB^{2} = A. Then the matrix AA is normal.

Problem 14.1.19

We know that all real symmetric matrices are diagonalizable. Are all complex symmetric matrices are diagonalizable?

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Problem 14.1.20

Let AA be a nonnormal n×nn \times n matrix and let UU be a unitary n×nn \times n matrix. Is A~=UAU−1\widetilde A = UAU^{-1} nonnormal for all UU?

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Problem 14.1.21

Let AA be a nonnormal matrix. Show that A∗AA^{*}A and AA∗AA^{*} are linearly independent.

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Problem 14.1.22
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(i)

Show that if AA and BB are nonnormal, then A⊗BA \otimes B is nonnormal.

(ii)

Show that if AA and BB are nonnormal, then A⊕BA \oplus B is nonnormal, where ⊕\oplus denotes the direct sum.

Section 14.2Supplementary Problems[13/0%]
Problem 14.2.1

Are all nonzero nilpotent matrices nonnormal?

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Problem 14.2.2
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(i)

Let a11,a22,ϵ∈Ra_{11}, a_{22}, \epsilon \in \mathbb {R}. What is the condition on a11a_{11}, a22a_{22}, ϵ\epsilon such that the 2×22 \times 2 matrix

A=(a11eϵe−ϵa22) A = \begin{pmatrix} a_{11} & e^{\epsilon } \\ e^{-\epsilon } & a_{22} \end{pmatrix}

is normal?

(ii)

Let a>0a > 0, b≥0b \geq 0 and ϕ∈[0,π]\phi \in [0,\pi ]. What are the conditions aa, bb, ϕ\phi such that

B(a,b,ϕ)=(0aeiϕb0) B(a,b,\phi ) = \begin{pmatrix} 0 & a \\ e^{i\phi }b & 0 \end{pmatrix}

is a normal matrix?

(iii)

Let z∈Cz \in \mathbb {C}. Is the 2×22 \times 2 matrix

(101−eze−z) \begin{pmatrix} 1 & 0 \\ 1-e^{z} & e^{-z} \end{pmatrix}

nonnormal?

Problem 14.2.3
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(i)

Let x1,x2,x3∈Rx_{1}, x_{2}, x_{3} \in \mathbb {R}. What is the condition such that the 3×33 \times 3 matrix

A(x1,x2,x3)=(0x1000x2x300) A(x_{1},x_{2},x_{3}) = \begin{pmatrix} 0 & x_{1} & 0 \\ 0 & 0 & x_{2} \\ x_{3} & 0 & 0 \end{pmatrix}

is normal? Show that λ3=x1x2x3\lambda^{3} = x_{1}x_{2}x_{3} for det⁡(A−λI3)=0\operatorname {det}\left(A-\lambda I_{3}\right) = 0. Find the eigenvalues and normalized eigenvectors of AA.

(ii)

Let x∈Rx \in \mathbb {R}. For which values of xx is the 3×33 \times 3 matrix

A(x)=(−1+ixixixixixixixix1+ix) A(x) = \begin{pmatrix} -1+ix & ix & ix \\ ix & ix & ix \\ ix & ix & 1+ix \end{pmatrix}

nonnormal.

Problem 14.2.4

Show that if AA is nonnormal, then A∙A∗A \bullet A^{*} is normal, where ∙\bullet denotes the Hadamard product. Notice that A∙B=B∙AA \bullet B = B \bullet A and (A∙B)∗=A∗∙B∗(A \bullet B)^{*} = A^{*} \bullet B^{*}.

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Problem 14.2.5

Find all nonnormal 2×22 \times 2 matrices AA such that AA∗+A∗A=I2AA^{*}+A^{*}A = I_{2}. An example is

A=(0100)⇒A∗=(0010). A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \Rightarrow A^{*} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.
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Problem 14.2.6
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(i)

Can one find a nonnormal matrix AA such that eAeA∗=eA∗eAe^{A}e^{A^{*}} = e^{A^{*}}e^{A}?

(ii)

Can one find a nonnormal matrix AA such that eAeA∗=eA+A∗e^{A}e^{A^{*}} = e^{A+A^{*}}?

Problem 14.2.7

Let MM be an invertible n×nn \times n matrix. Then we can form

M+=12(M+M−1),M−=12(M−M−1). M_{+} = \frac{1}{2}(M+M^{-1}), \quad M_{-} = \frac{1}{2}(M-M^{-1}).

Consider the nonnormal matrix

M=(1101). M = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}.

Find M+M_{+} and M−M_{-}. Are the matrices M+M_{+} and M−M_{-} normal?

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Problem 14.2.8

Let sj:=2sin⁡(2πj/5)s_{j} := 2\sin (2\pi j/5) with j=1,…,5j = 1, \ldots , 5. Consider the 5×55 \times 5 matrix

M=(s1100−1−1s21000−1s31000−1s41100−1s5). M = \begin{pmatrix} s_{1} & 1 & 0 & 0 & -1 \\ -1 & s_{2} & 1 & 0 & 0 \\ 0 & -1 & s_{3} & 1 & 0 \\ 0 & 0 & -1 & s_{4} & 1 \\ 1 & 0 & 0 & -1 & s_{5} \end{pmatrix}.

Show that the matrix is nonnormal. Find the eigenvalues and eigenvectors. Is the matrix diagonalizable?

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Problem 14.2.9

Let AA, BB be nonzero n×nn \times n hermitian matrices. We form the matrix KK as K:=A+iBK := A+iB. This matrix is nonnormal if [A,B]≠0n[A,B] \neq 0_{n} and normal if AB=BAAB = BA. In the following we assume that KK is nonnormal. Assume that we can find a unitary n×nn \times n matrix UU such that UKU∗=K∗UKU^{*} = K^{*}. What can be said about the eigenvalues of KK? In physics such a unitary matrix is called a quasi-symmetry operator.

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Problem 14.2.10

Let QQ be a nonnormal invertible n×nn \times n matrix. Is Q⊗Q−1Q \otimes Q^{-1} nonnormal?

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Problem 14.2.11

Let MM be an n×nn \times n nonnormal matrix. Are the 2n×2n2n \times 2n matrices

A=(0nM∗M0n),B=(0n−M∗M0n) A = \begin{pmatrix} 0_{n} & M^{*} \\ M & 0_{n} \end{pmatrix}, \quad B = \begin{pmatrix} 0_{n} & -M^{*} \\ M & 0_{n} \end{pmatrix}

nonnormal?

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Problem 14.2.12

Let AA be a nonnormal invertible matrix. How do we construct a matrix BB such that A=exp(B)A = \mathrm{exp}\left(B\right)? Study first the two examples

A1=(1101),A2=(2101/2). A_{1} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad A_{2} = \begin{pmatrix} 2 & 1 \\ 0 & 1/2 \end{pmatrix}.

Both are elements of the Lie group SL(2,R)SL(2,\mathbb {R}). The matrix A1A_{1} admits only one linearly independent eigenvector, namely (1,0)T(1,0)^{T}. The matrix

B1=(0100) B_{1} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}

generates A1A_{1}.

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Problem 14.2.13

Let AA be an n×nn \times n nonnormal matrix and Q∈SL(n,C)Q \in SL(n,\mathbb {C}). Can one find QQ such that A~=QAQ−1\widetilde A = QAQ^{-1} is normal?

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