Math Problems Vault

Problem 27.10

In a complex arithmetic space with the standard scalar product, find the scalar products of the vectors

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(1)

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

(2)

$\left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right], \left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1+2i \\ -1+2i \end{smallmatrix}\right], \left[\begin{smallmatrix} 2-i \\ 2+i \end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 \\ i \\ 1 \end{smallmatrix}\right], \left[\begin{smallmatrix} i \\ 1 \\ i \end{smallmatrix}\right]$

(5)

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

(6)

$\mathbf{c}_{222}, \mathbf{c}_{223}$ . $\mathbf{c}_{222} = \left[\begin{smallmatrix} 0 \\ 2 \\ 0 \\ 5-2 i \end{smallmatrix}\right], \mathbf{c}_{223} = \left[\begin{smallmatrix} 0 \\ 2+i \\ 0 \\ 6+i \end{smallmatrix}\right]$ .

(7)

$\mathbf{c}_{221}; \mathbf{c}_{215}$ . $\mathbf{c}_{221} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 3-i \end{smallmatrix}\right], \mathbf{c}_{215} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 - i \end{smallmatrix}\right]$ .

Problem 27.11

In problem 27. 10, for each pair of vectors find the length of the first vector:

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(1)

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

(2)

$\left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right], \left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1+2i \\ -1+2i \end{smallmatrix}\right], \left[\begin{smallmatrix} 2-i \\ 2+i \end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 1 \\ i \\ 1 \end{smallmatrix}\right], \left[\begin{smallmatrix} i \\ 1 \\ i \end{smallmatrix}\right]$

(5)

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

(6)

$\mathbf{c}_{222}, \mathbf{c}_{223}$ . $\mathbf{c}_{222} = \left[\begin{smallmatrix} 0 \\ 2 \\ 0 \\ 5-2 i \end{smallmatrix}\right], \mathbf{c}_{223} = \left[\begin{smallmatrix} 0 \\ 2+i \\ 0 \\ 6+i \end{smallmatrix}\right]$ .

(7)

$\mathbf{c}_{221}; \mathbf{c}_{215}$ . $\mathbf{c}_{221} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 3-i \end{smallmatrix}\right], \mathbf{c}_{215} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 - i \end{smallmatrix}\right]$ .

Problem 27.12

Find the scalar product of vectors of a unitary space from their coordinates in a basis $e$ and the Gram matrix $\Gamma$ of this basis:

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(1)

$\left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 1 \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 2 & i \\ - i & 2\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1+i \\ i \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 1-i \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 3 & 1 + i \\ 1 - i & 1\end{smallmatrix}\right]$

(3)

$\mathbf{c}_{40}, \mathbf{c}_{41}, \quad \Gamma = \mathbf{A}_{87}$ . $\mathbf{c}_{40} = \left[\begin{smallmatrix} 1+i \\ 3 \end{smallmatrix}\right], \mathbf{c}_{41} = \left[\begin{smallmatrix} 1+2 i \\ 6+2 i \end{smallmatrix}\right], \mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{44}, \quad \mathbf{c}_{40}, \quad \Gamma = \mathbf{A}_{86}$ . $\mathbf{c}_{44} = \left[\begin{smallmatrix} i \\ 2 \end{smallmatrix}\right], \mathbf{c}_{40} = \left[\begin{smallmatrix} 1+i \\ 3 \end{smallmatrix}\right], \mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(5)

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

(6)

$\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 2 & i & 0 \\ - i & 2 & - i \\ 0 & i & 2\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} -1 \\ 2+i \\ 1 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ -i \\ 1+i \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 1 & 1 - i & 0 \\ 1 + i & 3 & i \\ 0 & - i & 2\end{smallmatrix}\right]$

(8)

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

Problem 27.13

In problem 27. 12, for each pair of vectors find the length of the first vector:

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(1)

$\left[\begin{smallmatrix} 1 \\ i \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 2i \\ 1 \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 2 & i \\ - i & 2\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1+i \\ i \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 1-i \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 3 & 1 + i \\ 1 - i & 1\end{smallmatrix}\right]$

(3)

$\mathbf{c}_{40}, \mathbf{c}_{41}, \quad \Gamma = \mathbf{A}_{87}$ . $\mathbf{c}_{40} = \left[\begin{smallmatrix} 1+i \\ 3 \end{smallmatrix}\right], \mathbf{c}_{41} = \left[\begin{smallmatrix} 1+2 i \\ 6+2 i \end{smallmatrix}\right], \mathbf{A}_{87} = \left[\begin{smallmatrix} 1 & 1+i \\ 1-i & 3 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{44}, \quad \mathbf{c}_{40}, \quad \Gamma = \mathbf{A}_{86}$ . $\mathbf{c}_{44} = \left[\begin{smallmatrix} i \\ 2 \end{smallmatrix}\right], \mathbf{c}_{40} = \left[\begin{smallmatrix} 1+i \\ 3 \end{smallmatrix}\right], \mathbf{A}_{86} = \left[\begin{smallmatrix} 5 & i \\ -i & 1 \end{smallmatrix}\right]$ .

(5)

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

(6)

$\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ 0 \\ 1 \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 2 & i & 0 \\ - i & 2 & - i \\ 0 & i & 2\end{smallmatrix}\right]$

(7)

$\left[\begin{smallmatrix} -1 \\ 2+i \\ 1 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 \\ -i \\ 1+i \end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 1 & 1 - i & 0 \\ 1 + i & 3 & i \\ 0 & - i & 2\end{smallmatrix}\right]$

(8)

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

Problem 27.14

Prove that a Hermitian matrix $\Gamma$ can serve as the Gram matrix in a unitary space if and only if $\xi^{T}\Gamma \bar{\xi } > 0$ for every nonzero column $\xi$ .

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Problem 27.15

Prove that a square matrix $\Gamma$ of order $n$ can serve as the Gram matrix in an $n$ -dimensional unitary space if and only if there exists a nondegenerate matrix $S$ such that $\Gamma = S^{T} \bar{S}$ .

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Problem 27.16

From the bank, select the Hermitian matrices (excluding the real symmetric ones) and, among them, those that can serve as Gram matrices in a unitary space.

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Problem 27.17

In the Gram matrix of some basis in a unitary space, all nonzero elements are equal to 1 in modulus. Prove that the basis is orthonormal.

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Problem 27.18

(1)(2)(3)(4)

In the Gram matrix of some basis:

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(1)

a real number $\alpha$ was added to all elements of the main diagonal

(2)

a positive number $\alpha$ was added to some elements of the main diagonal

(3)

two rows were permuted

(4)

two rows were permuted, and also two columns with the same numbers as the rows.

Did the matrix remain Hermitian, and can it be the Gram matrix of some basis?

Problem 27.19

Prove that the determinant of the matrix formed from the pairwise scalar products of some system of vectors is a real nonnegative number, equal to zero if and only if this system of vectors is linearly dependent.

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Problem 27.20

Prove that for any complex matrix $\operatorname {Rg} A^{T} \bar{A} = \operatorname {Rg} A$ .

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Problem 27.21

From the bank, select the unitary matrices that are not real orthogonal matrices.

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Problem 27.22

Write down some unitary matrix of order three.

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Problem 27.23

Describe all triangular unitary matrices.

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Problem 27.24

(1)(2)(3)

Can a unitary matrix of order four contain the row:

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(1)

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

(2)

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

(3)

$\left[\begin{smallmatrix} \frac{1 + i}{\sqrt{8}} & \frac{1 - i}{\sqrt{8}} & \frac{ - 1 + i}{\sqrt{8}} & \frac{ - 1 - i}{\sqrt{8}} \end{smallmatrix}\right]$ .

Problem 27.25

(1)(2)

Let $A$ and $B$ be real square matrices of order $n$ . From them one can form a complex matrix $C = A + iB$ and a real matrix of order $2n$

D =\left[\begin{smallmatrix} [c|c] A & -B \\ \hline B & A \end{smallmatrix}\right]^{\square }

Prove that:

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(1)

$C$ is Hermitian if and only if $D$ is symmetric

(2)

$C$ is unitary if and only if $D$ is orthogonal.