Math Problems Vault

Problem 30.13

Find the condition on the matrix of a transformation $\varphi$ in an orthonormal basis, necessary and sufficient for $\varphi$ to be normal.

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Problem 30.14

(1)(2)(3)

For a normal transformation $\varphi$ of the space $\mathcal{U}$ , prove that:

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(1)

$\operatorname {Ker} \varphi = \operatorname {Ker} \varphi^{ * }$

(2)

$\operatorname {Im} \varphi = \operatorname {Im} \varphi^{ * }$ ; $\quad$

(3)

$\mathcal{U} = \operatorname {Ker} \varphi \oplus \operatorname {Im} \varphi$ .

Problem 30.15

Prove that a transformation $\varphi$ is normal if and only if every eigenvector of $\varphi$ is also an eigenvector of $\varphi^{ * }$ .

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Problem 30.16

Let $\mathcal{L}$ be an eigenspace of a normal transformation $\varphi$ . Prove that $\mathcal{L}^{\perp }$ is invariant under $\varphi$ .

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Problem 30.17

Let $x$ and $y$ be eigenvectors of a normal transformation $\varphi$ belonging to distinct eigenvalues. Prove that $x$ and $y$ are orthogonal.

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Problem 30.18

(1)(2)

Let $\varphi$ be a normal transformation of a unitary space $\mathcal{U}$ . Prove that

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(1)

$\mathcal{U}$ is the direct sum of pairwise orthogonal eigenspaces of the transformation $\varphi$ .

(2)

In $\mathcal{U}$ there exists an orthonormal basis of eigenvectors of $\varphi$ .

Problem 30.19

Let there exist for a transformation an orthonormal basis of eigenvectors. Prove that it is normal.

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Problem 30.20

Prove that the product of normal transformations is normal if they commute. Is the converse true?

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Problem 30.21

Prove that a transformation $\varphi$ of a unitary space is normal if and only if for every invariant subspace $\mathcal{L}$ the subspace $\mathcal{L}^{\perp }$ is also invariant.

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Problem 30.22

(1)(2)(3)

A normal transformation is given in an orthonormal basis by the matrix $A$ . Find the transition matrix $S$ to an orthonormal basis of eigenvectors and the matrix $A^{\prime }$ of the transformation in this basis:

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(1)

$A =\left[\begin{smallmatrix} 2 & i \\ i & 2\end{smallmatrix}\right]$

(2)

$A =\left[\begin{smallmatrix} 0 & 0 & - 3 \\ 0 & 0 & 4 \\ 3 & - 4 & 0\end{smallmatrix}\right]$

(3)

$A = \frac{1}{3}\left[\begin{smallmatrix} 3 - i & 2i & 0 \\ 2i & 3 & 2i \\ 0 & 2i & 3 + i\end{smallmatrix}\right]$ .

Problem 30.23

Let the transformations $\varphi$ and $\psi$ be normal, and $\varphi \psi = o$ . Does it follow that $\psi \varphi = o$ ?

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Problem 30.24

(1)(2)

Prove that:

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(1)

for any complex matrix, the sum of the squares of the moduli of all elements is not less than the sum of the squares of the moduli of all eigenvalues (each counted with its multiplicity)

(2)

for a normal matrix, the sums mentioned in the first item are equal.