COMPLEX MATRICES

So far, we have considered matrices whose elements were real numbers. The elements
of a matrix can, however, be complex numbers also.

\textbf{(1) Conjugate of a matrix.} If the elements of a matrix \(A = [a_{rs}]\)

are complex numbers \(\alpha_{rs} + i\beta_{rs}\),\(\alpha_{rs}\), \(\beta_{rs}\)
are real, then the matrix

\[
\overline{A} = [\overline{a_{rs}}] = [\alpha_{rs} - i\beta_{rs}]
\]
is called the conjugate matrix of \(A\).

The transpose of a conjugate of a matrix \(A\) is denoted by \(A^{*}\) or \(A^{\

theta}\), i.e.,

\[
(\overline{A})^- = A^{*}.
\]

\textbf{(2) Hermitian matrix.} A square matrix \(A\) such that \(A' = \

overline{A}\) is said to be a Hermitian matrix. The elements of the leading
diagonal of a Hermitian matrix are evidently real, while every other element is the
complex conjugate of the element in the transposed position. For instance,

\[
A = \begin{bmatrix}
2 & 3 + 4i \\
3 - 4i & -5
\end{bmatrix}
\]

is a Hermitian matrix, since

\[
A' = \begin{bmatrix}
2 & 3 - 4i \\
3 + 4i & -5
\end{bmatrix} = \overline{A}.
\]

\textbf{(3) Skew-Hermitian matrix.} A square matrix \(A\) such that \(A' = -\

overline{A}\) is said to be a skew-Hermitian matrix. This implies that the leading
diagonal elements of a skew-Hermitian matrix are either all zeros or all purely
imaginary.

\textbf{Observation:} A Hermitian matrix is a generalization of a real symmetric

matrix as every real symmetric matrix is Hermitian. Similarly, a skew-Hermitian
matrix is a generalization of a real skew-symmetric matrix.

\textbf{Properties}

\textbf{I.} Any square matrix \(A\) can be written as the sum of a Hermitian and
skew-Hermitian matrices.

Take
\[
B = \frac{1}{2}(A + \overline{A}')
\]
and
\[
C = \frac{1}{2}(A - \overline{A}').
\]

Then

\[
B' = \frac{1}{2}(A + \overline{A})' = \frac{1}{2}(A' + \overline{A})
\]

and

\[
\overline{B} = \frac{1}{2}\overline{(A + \overline{A}')} = \frac{1}{2}(\overline{A}
+ A') = B'.
\]

Therefore, \(B\) is a Hermitian matrix.

Again,

\[
C' = \frac{1}{2}(A - \overline{A}')' = \frac{1}{2}(A' - \overline{A})
\]

and

\[
\overline{C} = \frac{1}{2}\overline{(A - \overline{A}')} = \frac{1}{2}(\overline{A}
- A') = -C'.
\]

Thus, \(C' = -C\), i.e., \(C\) is a skew-Hermitian matrix.

\[
A = \frac{1}{2}(A+ \overline{A}') = \frac{1}{2}(A - \overline{A}')=B+C
\]
Hence the result.

\textbf{II.} If \(A\) is a Hermitian matrix, then \(iA\) is a skew-Hermitian

matrix.

We have

\[
(\overline{iA})' = ( \overline{i}\overline{A})' =( -\overline{i}\overline{A})' = -
i\overline{A}' = -iA
\]

Thus, \(iA\) is a skew-Hermitian matrix.

Similarly if a is a skew-Hermitian matrix then (iA) is Hermitian matrix.

III. The eigenvalues of a Hermitian matrix are real.

Let \(\lambda\) be the eigenvalue and \(X\) the corresponding eigen vector of a
Hermitian matrix \(A\), so that
\[
AX = \lambda X.
\] \[
\overline{X}'AX=\overline{X}'\lambda X = \lambda \overline{X}'X.
\]

Since \(\overline{X}'X\) is Hermitian, \(\overline{x_1}x_1+\overline{x_2}x_2+...+\

overline{x_n}x_n=|x_1|^2+|x_2|^2+....+|x_n|^2\) is real and non-zero. Also \[\
overline{X}'AX\] is a Hermitian form which is always real.
$\therefore \lambda$, the eigen value of a Hermitian matrix is real.

IV. The eigen values of a skew-Hermitian matrix are purely imaginary or zero.

Let λ be the eigen value and X the corresponding eigen vector of a skew-Hermitian
matrix B so that $BX = λX.$

$X'BX = X'λX = λX'X or λ = X'BX/X'X$

Since $X'X$ is real and non-zero. Also $X'BX$ is a skew-Hermitian form which is
purely imaginary or zero.

∴ λ, the eigen value of a skew-Hermitian matrix is purely imaginary or zero.

\textbf{4. }Unitary Matrix: A square matrix \(U\) such that \(\overline{U}'=U^{-

1} \) is called a unitary matrix. For a unitary matrix \(U\), we have
\[
UU^* = U^*U = I.
\]
This is a generalization of the orthogonal matrix in the complex field.

\subsection{Properties}

\begin{enumerate}
\item \textbf{Inverse of a unitary matrix is unitary}
If \(U\) is a unitary matrix, then
\[
\overline{U}'=U^{-1}
\]
and thus
\[ U'= \overline{U^{-1}} . \]
\[ \therefore [(U^{-1)-1}]'= \overline{U^{-1}} \]
Let \(U^{-1} = V\). We have
\[
[V^{-1}]' = \overline{V} , \quad V^{-1}=\overline{V}' \]

Thus, \(V\) (which is \(U^{-1}\)) is also unitary. Consequently, the inverse of

an orthogonal matrix is orthogonal.

\item \textbf{Transpose of a unitary matrix is unitary}

If \(U\) is a unitary matrix,
\[\overline{U}'= U^{-1}
\]
and
\[
[ (\overline{U'})]'= [U^{-1}]'=[U']^{-1}
\]
Therefore, \(U'=V\), is we have $\overline{V'}=V^{-1}$
 Thus V is also unitary.

Consequently, the transpose of an orthogonal matrix is orthogonal.

\item \textbf{Product of two unitary matrices is a unitary matrix}

If \(U\) and \(V\) are unitary matrices, then

\[
U'=\overline{U}^{-1}, V' = \overline{V}^{-1}
\]
Thus,
\[
\overline{UV}^{-1}=\overline{UV}^{-1}=\overline{V}^{-1}\overline{U}^{-1} =V'U' =
(UV)'
\]
Thus, \(UV\) is a unitary matrix.

\textbf{Corollary:} The product of two orthogonal matrices is an orthogonal matrix.

4. Eigenvalues of a Unitary Matrix

The eigenvalue of a unitary matrix has absolute value 1.

If \(U\) is a unitary matrix, then

\[
UX = \lambda X.
\]

Taking the conjugate transpose of this equation,

\[(\overline{UX}') = (\overline{UX}') = \overline{X'} \overline{U'} = \

overline{X}'\overline{U}^{-1} \] \[
(\overline{UX}') = (\overline{\lambda X})' = \overline{\lambda} \
overline{X'} \]Also \[
\overline{X'}U^{-1} = \overline{\lambda}\overline{X'} \]
Post multiplying (2) by (1), we get
\[(\overline{X'}U^{-1})(UX) = (\overline{\lambda}\overline{X'})=(\lambda X) \]
\[\overline{X'}(U^{-1}U)X = (\overline{\lambda}\lambda)(\overline{X'}X )\]
\[\overline{X'}X = (\lambda\lambda')\overline{X'}X \]
\[ \lambda\lambda' = |\lambda|^2 = 1\]
Hence the result.

\textbf{Corollary:} The eigenvalue of an orthogonal matrix has absolute value 1.

give 10 more questions for the same topic with same format and please give
practical questions instead of definations and please dont repeat the questions

hints should be proper dont give exact solution in hint, give 3 hints for all the
questions, each hint should have order number and give four options for each
question and brief explaination for correct answer and the topic name should be
complex matrices

from the above topic give 10 programmatical questions in the below format
{
"content": "Determine whether \\(A = \\begin{bmatrix} 2 & 4 & 6 \\
end{bmatrix}\\) is a row matrix.",
"type": "multiple_choice",
"tags": [
"row matrix", "matrix structure"
],
"topics": [
"special matrices"
],
"taxonomyAspect": "Remember",
"complexityLevel": "Easy",
"hints": [
{
"title": "Hint 1",
"content": "Check how many rows are in the matrix.",
"order": 1
},
{
"title": "Hint 2",
"content": "Count the number of rows and columns in the matrix. A
row matrix has only one row.",
"order": 2
}
],
"options": [
{
 "content": "Yes",
"isCorrect": true,
"explanation": "Correct! Matrix \\(A = \\begin{bmatrix} 2 & 4 &
6 \\end{bmatrix}\\) has 1 row and 3 columns. A row matrix is defined as a matrix
with only one row, regardless of the number of columns. Since \\(A\\) has only one
row, it is indeed a row matrix."
},
{
"content": "No",
"isCorrect": false,
"explanation": "Incorrect. A matrix with a single row is a row
matrix, and \\(A\\) satisfies this condition."
},
{
"content": "Cannot determine",
"isCorrect": false,
"explanation": "Incorrect. The matrix clearly has only one row."
},
{
"content": "It is a column matrix",
"isCorrect": false,
"explanation": "Incorrect. A column matrix must have only one
column, but \\(A\\) has multiple columns."
}
]
},
hints should be proper dont give exact solution in hint, give 3 hints for all
the questions, each hint should have order number and give four options for each
question and brief explaination for correct answer and the topic name should be
complex matrices

"Correct! The operations were applied correctly:

explain this step by step solution for below question and add step by step
solution in explanation itself dont give it seperately and give in json format