No, a Hermitian matrix is not always symmetric. While they share some similarities, particularly when dealing with real numbers, their definitions differ fundamentally when complex numbers are involved.
Understanding Symmetric and Hermitian Matrices
To clarify their relationship, let's first define each type of matrix:
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Symmetric Matrix
A square matrix $A$ is called symmetric if it is equal to its transpose ($A = A^T$). This means that the element in the $i$-th row and $j$-th column is identical to the element in the $j$-th row and $i$-th column ($A{ij} = A{ji}$). Symmetric matrices are most commonly discussed in the context of real number entries.
- Example (Real Symmetric):
A = [ 1 2 ] [ 2 3 ]Here, $A^T = A$.
- Example (Real Symmetric):
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Hermitian Matrix
A square matrix $A$ is called Hermitian if it is equal to its conjugate transpose ($A = A^H$ or $A = A^*$). The conjugate transpose involves taking the transpose of the matrix and then taking the complex conjugate of each entry. Therefore, for a Hermitian matrix, $A{ij} = \overline{A{ji}}$, where $\overline{z}$ denotes the complex conjugate of $z$. Hermitian matrices are fundamental in fields like quantum mechanics and frequently involve complex numbers.
- Example (Complex Hermitian):
B = [ 1 2+i ] [ 2-i 3 ]Here, $\overline{B} = \begin{bmatrix} 1 & 2-i \ 2+i & 3 \end{bmatrix}$, and $B^H = (\overline{B})^T = \begin{bmatrix} 1 & 2+i \ 2-i & 3 \end{bmatrix}$. So, $B = B^H$.
- Example (Complex Hermitian):
The Key Difference: Transpose vs. Conjugate Transpose
The crucial distinction lies in how "transposition" is handled:
- For a symmetric matrix, we only consider the transpose ($A^T$).
- For a Hermitian matrix, we consider the conjugate transpose ($A^H$), which involves both transposition and complex conjugation.
This difference becomes significant when a matrix contains complex entries. If an entry is a non-real complex number (e.g., $2+i$), its complex conjugate ($2-i$) will be different from itself. Consequently, the conjugate transpose will generally differ from the simple transpose.
When a Hermitian Matrix IS Symmetric
A Hermitian matrix is symmetric only under a specific condition: when all its entries are real numbers.
Let's break down why:
- If a matrix $A$ is Hermitian, it must satisfy $A{ij} = \overline{A{ji}}$.
- For the same matrix $A$ to also be symmetric, it must satisfy $A{ij} = A{ji}$.
For a matrix to be both Hermitian and symmetric, both conditions must hold simultaneously for every element: $A{ij} = \overline{A{ji}}$ and $A{ij} = A{ji}$.
Combining these, we must have $A{ji} = \overline{A{ji}}$. For any complex number to be equal to its own conjugate, its imaginary part must be zero. Therefore, every entry in the matrix must be a real number.
Important Insight: It's worth noting the inverse relationship: Every real symmetric matrix is Hermitian. If a matrix has only real entries and is symmetric ($A = A^T$), then its complex conjugate ($\overline{A}$) is just the matrix itself ($A$), because real numbers are their own conjugates. Therefore, its conjugate transpose $A^H = (\overline{A})^T = A^T$. Since $A=A^T$, it follows that $A=A^H$, making it Hermitian. This highlights the specific case where the two definitions overlap.
Illustrative Examples
Let's look at examples to solidify this concept:
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A Matrix that is Both Symmetric and Hermitian (Real Symmetric):
Consider matrix $A$:
A = [ 1 5 ] [ 5 7 ]- Symmetric Check: $A^T = \begin{bmatrix} 1 & 5 \ 5 & 7 \end{bmatrix}$. Since $A^T = A$, $A$ is symmetric.
- Hermitian Check: Since all entries are real, $\overline{A} = A$. Thus, $A^H = (\overline{A})^T = A^T = A$. Since $A^H = A$, $A$ is Hermitian.
- Conclusion: This matrix is both symmetric and Hermitian.
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A Matrix that is Hermitian but NOT Symmetric (Complex Hermitian):
Consider matrix $B$:
B = [ 1 2 + 3i ] [ 2 - 3i 4 ]- Symmetric Check: $B^T = \begin{bmatrix} 1 & 2 - 3i \ 2 + 3i & 4 \end{bmatrix}$. Clearly, $B \neq B^T$ because $2+3i \neq 2-3i$. Thus, $B$ is not symmetric.
- Hermitian Check:
First, find the complex conjugate of $B$:
$\overline{B} = \begin{bmatrix} 1 & 2 - 3i \ 2 + 3i & 4 \end{bmatrix}$
Then, find the transpose of $\overline{B}$ (the conjugate transpose $B^H$):
$B^H = (\overline{B})^T = \begin{bmatrix} 1 & 2 + 3i \ 2 - 3i & 4 \end{bmatrix}$.
Since $B^H = B$, $B$ is Hermitian. - Conclusion: This matrix is Hermitian but not symmetric.
Summary Table: Symmetric vs. Hermitian
Here's a quick comparison of the key properties:
| Feature | Symmetric Matrix | Hermitian Matrix |
|---|---|---|
| Definition | $A = A^T$ (Matrix equals its transpose) | $A = A^H$ (Matrix equals its conjugate transpose) |
| Element Relation | $A{ij} = A{ji}$ | $A{ij} = \overline{A{ji}}$ |
| Entry Type | Typically real; can be complex if $A{ij} = A{ji}$ | Can be real or complex numbers |
| Real Case | Every real symmetric matrix is Hermitian | Every real Hermitian matrix is symmetric |
| Complex Case | A complex symmetric matrix is generally not Hermitian | A complex Hermitian matrix is generally not symmetric |
| Diagonal Entries | Can be complex | Must be real (since $A{ii} = \overline{A{ii}}$) |
| Eigenvalues | Real (if entries are real) | Always real |
Practical Significance
The distinction between symmetric and Hermitian matrices is not just theoretical; it has profound implications in various scientific and engineering fields:
- Quantum Mechanics: Hermitian operators (represented by Hermitian matrices) correspond to observable physical quantities like position, momentum, and energy. Their eigenvalues, which represent the possible measurement outcomes, are always real numbers, consistent with physical reality.
- Linear Algebra and Numerical Analysis: Both types of matrices exhibit desirable properties, such as having real eigenvalues and orthogonal (or unitary) eigenvectors. These properties simplify many computations and are essential for stability analysis in numerical methods.
- Engineering and Physics: Symmetric matrices commonly appear in stress-strain tensors, moment of inertia tensors, and adjacency matrices of undirected graphs, reflecting inherent symmetries in physical systems.
Conclusion
In summary, a Hermitian matrix is not universally symmetric. These two types of matrices are equivalent only in the special case where all entries of the matrix are real numbers. If a Hermitian matrix contains any complex numbers with non-zero imaginary parts, it will satisfy the Hermitian condition ($A = A^H$) but will fail the symmetric condition ($A = A^T$). The broader definition of Hermitian matrices is crucial for applications involving complex numbers, such as quantum mechanics, where the conjugate transpose naturally extends the concept of symmetry from real to complex vector spaces.
