Linear transformations of a unitary space

the matrix $P$ of orthogonal projection onto the subspace $\mathcal{L}$,

the matrix $Q$ of reflection in the subspace $\mathcal{L}$.

$\left[\begin{smallmatrix} 1 \\ i \\ 1 + i\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} i \\ 1 \\ 1-i\end{smallmatrix}\right]$;

$\left[\begin{smallmatrix} i \\ i \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 0 \\ 3\end{smallmatrix}\right]$;

$\left[\begin{smallmatrix} i \\ 1 \\ 1 \\ i\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 0 \\ 3 \\ i\end{smallmatrix}\right]$.

$\mathbf{a} = \left[\begin{smallmatrix} 3 \\ -1 \\ 2i \end{smallmatrix}\right]$

$\mathbf{a} = \mathbf{c}_{153}$. $\mathbf{c}_{153} = \left[\begin{smallmatrix} -i \\ -1 \\ -3 i \end{smallmatrix}\right]$.

$\mathbf{a} = \mathbf{c}_{131}$. $\mathbf{c}_{131} = \left[\begin{smallmatrix} 1 \\ -i \\ 1+i \end{smallmatrix}\right]$.

$\mathbf{a} = \mathbf{c}_{215}$. $\mathbf{c}_{215} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 - i \end{smallmatrix}\right]$.

$(\varphi + \psi )^{ * } = \varphi^{ * } + \psi^{ * }$;

$(\varphi \psi )^{ * } = \psi^{ * } \varphi^{ * }$;

$(\alpha \varphi )^{ * } = \bar{\alpha } \varphi^{ * }$;

if $\varphi$ has an inverse, then $\varphi^{ * }$ also has an inverse, and $\left(\varphi^{ * }\right)^{ - 1} = \left(\varphi^{ - 1}\right)^{ * }$.

$\mathbf{A}_{88}$. $\mathbf{A}_{88} = \left[\begin{smallmatrix} 1 & i \\ i & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{103}$. $\mathbf{A}_{103} = \left[\begin{smallmatrix} 0 & i \\ -i & 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{100}$. $\mathbf{A}_{100} = \left[\begin{smallmatrix} 1 & 2 i \\ i & -1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{92}, \quad \Gamma = \mathbf{A}_{87}$. $\mathbf{A}_{92} = \left[\begin{smallmatrix} i & 1 \\ 1 & i \end{smallmatrix}\right], \mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{97}, \quad \Gamma = \mathbf{A}_{86}$. $\mathbf{A}_{97} = \left[\begin{smallmatrix} i & 1 \\ -1 & 2 i \end{smallmatrix}\right], \mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{375}, \quad \Gamma = \mathbf{A}_{388}$. $\mathbf{A}_{375} = \left[\begin{smallmatrix} 1 & 1+i & 1-i \\ 1 & 1 & 1 \\ -1 & -i & i \end{smallmatrix}\right], \mathbf{A}_{388} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{smallmatrix}\right]$.

Let a vector $x$ be an eigenvector for the transformation $\varphi$ with eigenvalue $\lambda$ and an eigenvector for $\varphi^{ * }$ with eigenvalue $\mu$. Prove that $\lambda = \bar{\mu }$.

Prove that the transformation $\varphi^{ * }$, adjoint to a transformation $\varphi$ with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$, has eigenvalues $\overline{\lambda_{1}}, \ldots , \overline{\lambda_{n}}$.

Prove that the set of values of a linear transformation $\varphi$ of a unitary space coincides with the orthogonal complement of the kernel of the adjoint transformation $\varphi^{ * }$.

Prove that the Fredholm theorem for complex systems of linear equations also holds in the following formulation: the system $A\mathbf{x} = \mathbf{b}$ is consistent if and only if every solution of the system of linear equations $\bar{A}^{T} \mathbf{y} = \mathbf{0}$ satisfies the equality $\mathbf{y}^{T} \bar{b} = 0$.

Prove that every linear transformation of a unitary space has an $(n - 1)$-dimensional invariant subspace. Is this statement true for complex vector spaces?

Prove that for every linear transformation of a unitary space there exists an orthonormal basis in which the matrix of the transformation is upper triangular.

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

$A =\left[\begin{smallmatrix} - 1 & - 1 + i & - 1 - i \\ 1 + i & 1 & 1 + i \\ 1 - i & 1 - i & 1\end{smallmatrix}\right]$

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

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

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

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

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

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

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

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

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

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)

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

self-adjoint transformations can be defined as normal transformations whose eigenvalues are real

unitary transformations can be defined as normal transformations whose eigenvalues have modulus 1.

every transformation $\varphi$ of a unitary space can be represented in the form $\varphi = \varphi_{1} + i\varphi_{2}$, where $\varphi_{1}$ and $\varphi_{2}$ are self-adjoint transformations

$\varphi_{1}$ and $\varphi_{2}$ commute if and only if $\varphi$ is a normal transformation.

$\varphi$ is self-adjoint

$\varphi$ satisfies the condition $\varphi = - \varphi^{ * }$.

self-adjoint

unitary.

$\left[\begin{smallmatrix} 7 & 3i \\ - 3i & - 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 4 & \sqrt{3} + i \\ \sqrt{3} - i & 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 5 & \sqrt{2}(1 + i) \\ \sqrt{2}(1 - i) & 2\end{smallmatrix}\right]$

$\mathbf{A}_{377}$. $\mathbf{A}_{377} = \left[\begin{smallmatrix} 1 & 1-i & 0 \\ 1+i & 3 & i \\ 0 & -i & 1 \end{smallmatrix}\right]$.

the transformation $\psi = (\varphi - i\iota )^{ - 1}(\varphi + i\iota )$ is defined and is unitary

$\psi - \iota$ has an inverse, and $\varphi = i(\psi + \iota )(\psi - \iota )^{ - 1}$.

Prove that a linear transformation $\varphi$ is unitary if and only if $\varphi^{ * } = \varphi^{ - 1}$.

Prove that for a unitary transformation the adjoint is also unitary.

carries some orthonormal basis into an orthonormal basis

preserves the pairwise scalar products of the basis vectors of some basis.

Will the sum of unitary transformations be a unitary transformation?

Will the product of a unitary transformation by a number $\alpha$ be a unitary transformation?

$A =\left[\begin{smallmatrix} \cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$

$A = \frac{1}{5}\left[\begin{smallmatrix} 3 + 3i & \sqrt{7} \\ - \sqrt{7} & 3 - 3i\end{smallmatrix}\right]$

$A = \frac{1}{9}\left[\begin{smallmatrix} 4 + 3i & 4i & - 6 - 2i \\ - 4i & 4 - 3i & - 2 - 6i \\ 6 + 2i & - 2 - 6i & 1\end{smallmatrix}\right]$.