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Chapter 8: Problem 6

Show that a triangular hermitian matrix is diagonal.

Short Answer

Expert verified
A triangular Hermitian matrix A satisfies two conditions: \(A_{ij} = 0\) for \(i > j\) and \(A_{ij} = \overline{A_{ji}}\) for all \(i\) and \(j\). By analyzing elements above and below the main diagonal, we showed that all off-diagonal elements are equal to their complex conjugates, which implies they must be real numbers. Since A is triangular, these real off-diagonal elements must be zeros. Therefore, a triangular Hermitian matrix is necessarily diagonal.

Step by step solution

01

Write down the conditions for a triangular Hermitian matrix

The given matrix, let's call it A, should satisfy two conditions: 1. It is upper triangular: \(A_{ij} = 0\) for \(i > j\). 2. It is Hermitian: \(A_{ij} = \overline{A_{ji}}\) for all \(i\) and \(j\). From these conditions, we will show that if A is both triangular and Hermitian, it must be diagonal, i.e., all its off-diagonal elements are zero.
02

Show that the off-diagonal elements are equal to their complex conjugates

Let's pick an element above the main diagonal, \(A_{ij}\), where \(i < j\). Since A is upper triangular, we know that \(A_{ji} = 0\). Also, because A is Hermitian, we know that \(A_{ij} = \overline{A_{ji}} = \overline{0} = 0\). Therefore, \(A_{ij}\) is equal to its complex conjugate. For elements below the main diagonal, we can pick an element \(A_{ij}\) where \(i > j\). In this case, since A is Hermitian, we have \(A_{ij} = \overline{A_{ji}}\). However, we know that \(A_{ij}\) is equal to zero since A is upper triangular. Thus, \(0 = \overline{A_{ji}}\), which means that \(A_{ji}\) is also equal to its complex conjugate.
03

Conclude that the matrix is diagonal

We have shown that all the off-diagonal elements are equal to their complex conjugates. The only numbers that are equal to their own complex conjugates are real numbers. Therefore, all the off-diagonal elements are real numbers. Since our matrix is triangular, these real off-diagonal elements must be zeros. This leaves us with a diagonal matrix. Hence we conclude that a triangular Hermitian matrix is necessarily diagonal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangular Matrix
A triangular matrix can take one of two forms: an upper triangular matrix or a lower triangular matrix. A matrix is called upper triangular if all the entries below the main diagonal are zero. Similarly, for a lower triangular matrix, all entries above the main diagonal are zero. These special forms of matrices simplify many mathematical operations. Triangular matrices have some interesting properties:
  • In an upper triangular matrix, non-zero elements are allowed only on or above the main diagonal.
  • In a lower triangular matrix, non-zero elements are allowed only on or below the main diagonal.
  • Triangular matrices can be directly upper triangular, lower triangular, or both (which implies they are diagonal).
Triangular matrices are particularly useful in solving systems of linear equations and simplifying matrix calculations. For instance, they can streamline the process of matrix multiplication and aid in determining eigenvectors and eigenvalues.
Diagonal Matrix
A diagonal matrix is a kind of triangular matrix where all off-diagonal elements are zero. This means that only the elements on the main diagonal (from top left to the bottom right) can be non-zero. Mathematically, a matrix is diagonal if all elements outside the main diagonal are zero 鈥� that is, for a matrix \( A \), \( A_{ij} = 0 \) whenever \( i eq j \).Diagonal matrices possess unique features:
  • They are a subset of both upper and lower triangular matrices.
  • When you multiply a diagonal matrix by another diagonal matrix, the result is also a diagonal matrix.
  • Inverse of a diagonal matrix (if it exists) is also diagonal with each diagonal element being the reciprocal of the original diagonal element.
They are incredibly simple to work with in calculations since operations like matrix addition, subtraction, and multiplication become straightforward as they primarily involve the diagonal elements. Diagonal matrices are pivotal in linear algebra because they simplify complex equations and calculations.
Complex Conjugates
In mathematics, particularly in complex number theory, each complex number has a counterpart known as its complex conjugate. Given a complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, its complex conjugate \( \overline{z} \) is defined as \( a - bi \).Complex conjugates are essential due to their properties:
  • The product of a complex number and its conjugate is always a non-negative real number.
  • Complex conjugates are used to simplify the division of complex numbers, as multiplying by the conjugate removes the imaginary part from the denominator.
  • They play a crucial role in ensuring matrices like Hermitian matrices are composed correctly, where certain symmetry conditions must be satisfied.
Understanding complex conjugates is crucial for working with Hermitian matrices, as Hermitian matrices require that each element is equal to the complex conjugate of its counterpart relative to the main diagonal. This property ensures all elements in a Hermitian matrix are complex conjugates of their diagonally opposite counterparts, playing a pivotal role in determining other matrix properties.

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Most popular questions from this chapter

Let \(V\) be a finite dimensional space over the field \(K\), with a non- degenerate scalar product. Let \(A: V \rightarrow V\) be a linear map. Show that the image of \(^{t} A\) is the orthogonal space to the kernel of \(A\).

Which of the following matrices are hermitian: (a) \(\left(\begin{array}{rr}2 & i \\ -i & 5\end{array}\right)\) (b) \(\left(\begin{array}{cc}1+i & 2 \\ 2 & 5 i\end{array}\right)\) (c) \(\left(\begin{array}{ccc}1 & 1+i & 5 \\ 1-i & 2 & i \\ 5 & -i & 7\end{array}\right)\)

(a) A matrix \(A\) is called skew-symmetric if ' \(A=-A\). Show that any matrix M can be expressed as a sum of a symmetric matrix and a skewsymmetric one, and that these latter are uniquely determined. [Hint: Let \(\left.A=\frac{1}{2}\left(M+{ }^{t} M\right) .\right]\) (b) If \(A\) is skew-symmetric then \(A^{2}\) is symmetric. (c) Let \(A\) be skew-symmetric. Show that \(\operatorname{Det}(A)\) is 0 if \(A\) is an \(n \times n\) matrix and \(n\) is odd.

Let \(A\) be a hermitian matrix. Show that \(^{t} A\) and \(\bar{A}\) are hermitian. If \(A\) is invertible, show that \(A^{-1}\) is hermitian.

Show that the absolute value of the determinant of a real unitary matrix is equal to 1. Conclude that if \(A\) is real unitary, then \(\operatorname{Det}(A)=1\) or \(-1\).
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