Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 16
Find all \(a, b \in \mathbb{C}\) such that the matrix \(\left[\begin{array}{rr}a & -i \\ i & b\end{array}\right]\) is unitarily diagonalizable.
Short Answer
Expert verified
For the matrix to be unitarily diagonalizable, \(a = x + yi\) and \(b = -yi + xi\).
Step by step solution
01
Understand Unitary Diagonalizability
A matrix is unitarily diagonalizable if there exists a unitary matrix \(U\) such that \(U^*AU\) is a diagonal matrix, where \(A\) is the given matrix and \(U^*\) is the conjugate transpose of \(U\). A necessary and sufficient condition for unitary diagonalizability is that \(A\) is a normal matrix, meaning \([A, A^*] = 0\).
02
Compute the Conjugate Transpose
The given matrix is \(A = \begin{bmatrix} a & -i \ i & b \end{bmatrix}\). Its conjugate transpose is \(A^* = \begin{bmatrix} \overline{a} & -i \ i & \overline{b} \end{bmatrix}\). Here, \(\overline{a}\) and \(\overline{b}\) are the complex conjugates of \(a\) and \(b\), respectively.
03
Set Up the Normal Matrix Condition
For the matrix \(A\) to be normal, the commutator \([A, A^*] = AA^* - A^*A\) should be zero. Compute \(AA^*\) and \(A^*A\) to check this condition.
04
Compute AA*
Calculate \(AA^* = \begin{bmatrix} a & -i \ i & b \end{bmatrix} \begin{bmatrix} \overline{a} & -i \ i & \overline{b} \end{bmatrix} = \begin{bmatrix} |a|^2 + 1 & -a i + i \overline{b} \ a i - i \overline{b} & 1 + |b|^2 \end{bmatrix}\).
05
Compute A*A
Calculate \(A^*A = \begin{bmatrix} \overline{a} & -i \ i & \overline{b} \end{bmatrix} \begin{bmatrix} a & -i \ i & b \end{bmatrix} = \begin{bmatrix} |a|^2 + 1 & -\overline{a} i + i b \ \overline{a} i - i b & 1 + |b|^2 \end{bmatrix}\).
06
Solve the Commutator Equation for Zero
Set \([A, A^*] = AA^* - A^*A = 0\), which results in two off-diagonal equations: \(-a i + i \overline{b} = -\overline{a} i + i b\) and \(a i - i \overline{b} = \overline{a} i - i b\). Simplify to obtain \(ai = b\) and \(\overline{b} = \overline{a}\).
07
Simplify and Find Conditions on a and b
From \(ai = b\), we have \(b = ai\). From \(\overline{a} = b\), we have \(b = \overline{a}\). Combining these, we determine \(a = x + yi\) with \(b = -yi + xi\), hence \(a = \overline{b}\). Therefore, any \(a\) with \(b = \overline{a}i\) satisfies these conditions.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Normal Matrix
A matrix is said to be normal if it commutes with its conjugate transpose. In mathematical terms, this means that for a matrix \( A \), the equation \([A, A^*] = AA^* - A^*A = 0\) must hold true. When a matrix is normal, it is unitarily diagonalizable, which means it can be transformed into a diagonal matrix using a unitary matrix.
Key features include:
Key features include:
- Normal matrices are necessarily diagonalizable by a unitary matrix \( U \), such that \( U^*AU \) is diagonal.
- This property simplifies many matrix computations, making normal matrices very important in fields like quantum mechanics.
Conjugate Transpose
The conjugate transpose of a matrix, also known as the Hermitian transpose, is obtained by taking the transpose of the original matrix and then taking the complex conjugate of each entry. For a given matrix \( A = \begin{bmatrix} a & -i \ i & b \end{bmatrix} \), the conjugate transpose is denoted as \( A^* = \begin{bmatrix} \overline{a} & i \ -i & \overline{b} \end{bmatrix} \).
Important aspects:
Important aspects:
- The conjugate transpose is important because it helps in defining normal matrices.
- It plays a key role in simplifying expressions in linear algebra involving complex numbers.
- In some contexts, \( A = A^* \) identifies Hermitian matrices, which have real eigenvalues.
Complex Numbers
Complex numbers extend real numbers and include an "imaginary" unit \( i \), where \( i^2 = -1 \). A complex number has the form \( a + bi \), with \( a \) being the real part and \( b \) the imaginary part. In the context of matrices, complex numbers come in handy for describing transformations that include rotations.
Key elements to remember:
Key elements to remember:
- They can be added, subtracted, multiplied, and divided like other numbers.
- Their geometric representation is as points or vectors in a 2-D plane, making them incredibly useful in describing oscillations and waveforms.
- Complex conjugates, which involve changing the sign of the imaginary part \( \overline{a + bi} = a - bi \), are used in matrix operations to maintain important properties like orthogonality.
Commutator Equation
The commutator equation \([A, A^*] = 0\) is a critical condition for unitary diagonalizability. It checks whether a matrix \( A \) commutes with its conjugate transpose \( A^* \). When this condition is satisfied, it indicates the matrix is normal.
Key insights include:
Key insights include:
- The equation \([A, A^*] = AA^* - A^*A = 0\) allows it to be unitarily diagonalizable, simplifying analyses involving such matrices.
- In practical terms, solving the commutator equation often reduces to balancing off-diagonal terms in equations.
- Being able to solve this efficiently is key to designing algorithms in computational mathematics and physics.
