Lab 13. Symmetric and Hermitian Matrices: Independent Study
This lab accompanies Chapter 13: Symmetric and Hermitian Matrices.
The goal is to connect symmetric/Hermitian structure with computation, geometry, energy, and data:
- symmetric and skew-symmetric decompositions;
- Hermitian matrices and conjugate transpose;
- spectral decomposition using orthonormal eigenvectors;
- quadratic and Hermitian forms;
- positive definite and positive semidefinite tests;
- Gram matrices and \(A^*A\);
- Cholesky factorization.
This is an independent-study lab. Each main question includes a worked solution and a similar practice question.
Python practice notebook
You may use the Jupyter notebook version for longer Python practice:
Interactive lab
Study guide and worked questions
Question 1. Symmetric and skew-symmetric decomposition
Let
\[ M=\begin{bmatrix}1&4\\2&3\end{bmatrix}. \]
Find the symmetric part and the skew-symmetric part of \(M\).
Solution
Use
\[ S=\frac{M+M^T}{2}, \qquad K=\frac{M-M^T}{2}. \]
Then
\[ S=\begin{bmatrix}1&3\\3&3\end{bmatrix}, \qquad K=\begin{bmatrix}0&1\\-1&0\end{bmatrix}. \]
Thus
\[ M=S+K. \]
Similar practice
For
\[ M=\begin{bmatrix}2&-1\\5&0\end{bmatrix}, \]
find \(S\) and \(K\).
Answer
\[ S=\begin{bmatrix}2&2\\2&0\end{bmatrix}, \qquad K=\begin{bmatrix}0&-3\\3&0\end{bmatrix}. \]
Question 2. Hermitian check
Determine whether
\[ H=\begin{bmatrix}2&1+i\\1-i&4\end{bmatrix} \]
is Hermitian.
Solution
The diagonal entries are real. The off-diagonal entries are complex conjugates:
\[ \overline{1+i}=1-i. \]
Therefore
\[ H^*=H. \]
So \(H\) is Hermitian.
Similar practice
Determine whether
\[ B=\begin{bmatrix}1&2+i\\2+i&3\end{bmatrix} \]
is Hermitian.
Answer
No. The lower-left entry should be \(2-i\), not \(2+i\).
Question 3. Spectral decomposition
Let
\[ A=\begin{bmatrix}3&1\\1&3\end{bmatrix}. \]
Find an orthogonal diagonalization.
Solution
The eigenvalues are
\[ \lambda_1=4, \qquad \lambda_2=2. \]
Corresponding unit eigenvectors are
\[ \vec{u}_1=\frac{1}{\sqrt2}\begin{bmatrix}1\\1\end{bmatrix}, \qquad \vec{u}_2=\frac{1}{\sqrt2}\begin{bmatrix}1\\-1\end{bmatrix}. \]
Thus
\[ Q=\frac{1}{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}, \qquad D=\begin{bmatrix}4&0\\0&2\end{bmatrix}, \]
and
\[ A=QDQ^T. \]
Similar practice
Find an orthogonal diagonalization of
\[ A=\begin{bmatrix}5&2\\2&5\end{bmatrix}. \]
Answer
Eigenvalues are \(7\) and \(3\), with the same orthonormal eigenvectors
\[ \frac{1}{\sqrt2}(1,1)^T, \qquad \frac{1}{\sqrt2}(1,-1)^T. \]
Question 4. Positive definite test by eigenvalues
Classify
\[ A=\begin{bmatrix}3&1\\1&3\end{bmatrix}. \]
Solution
The eigenvalues are \(4\) and \(2\), both positive. Therefore \(A\) is positive definite.
Similar practice
Classify
\[ B=\begin{bmatrix}1&2\\2&1\end{bmatrix}. \]
Answer
The eigenvalues are \(3\) and \(-1\). Since one eigenvalue is negative, \(B\) is indefinite.
Question 5. Sylvester’s criterion
Use Sylvester’s criterion to test whether
\[ A=\begin{bmatrix} 2&1&0\\ 1&3&1\\ 0&1&2 \end{bmatrix} \]
is positive definite.
Solution
The leading principal minors are
\[ \Delta_1=2>0, \]
\[ \Delta_2=\det\begin{bmatrix}2&1\\1&3\end{bmatrix}=5>0, \]
and
\[ \Delta_3=\det(A)=8>0. \]
All leading principal minors are positive. Therefore \(A\) is positive definite.
Similar practice
Test
\[ B=\begin{bmatrix}1&2\\2&1\end{bmatrix}. \]
Answer
The leading principal minors are \(1\) and \(-3\). Since the second is negative, \(B\) is not positive definite.
Question 6. Why \(A^TA\) is positive semidefinite
Let \(A\in\mathbb{R}^{m\times n}\). Prove that \(A^TA\) is positive semidefinite.
Solution
For any \(\vec{x}\in\mathbb{R}^n\),
\[ \vec{x}^{\,T}A^TA\vec{x}=(A\vec{x})^T(A\vec{x})=\|A\vec{x}\|^2\ge0. \]
Therefore \(A^TA\) is positive semidefinite.
Similar practice
When is \(A^TA\) positive definite?
Answer
It is positive definite when \(A\) has full column rank, because then \(A\vec{x}\ne0\) for every nonzero \(\vec{x}\).
Notebook submission suggestions
In your notebook, include:
- at least one symbolic computation by hand;
- at least one numerical computation in Python;
- at least one explanation of the geometry or energy interpretation;
- one example you create yourself;
- one AI companion reflection, checked by your own computation.