Questions tagged [hermitian-matrices]

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.

In linear algebra, a hermitian matrix is a square matrix that is equal to its conjugate transpose; this is occasionally known as the adjoint matrix as well. Formally, matrix $A$ is symmetric if $A^H=A$. Real symmetric matrices are de facto hermitian.

The sum and difference of two hermitian matrices is again hermitian, but this is not always true for the product: given hermitian matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB = BA$. So for integer $n$, $A^n$ is hermitian if $A$ is hermitian. If $A^{−1}$ exists, it is hermitian if and only if $A$ is hermitian.

An important fact about hermitian (symmetric) matrices is that they possess real eigenvalues. This can be seen by considering a finite-dimensional complex vector space equipped with the usual inner product.

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How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \det \left(A^2+B^2+C^2\right)$$ This problem is…
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Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors. The definition is: A $n$…
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Matrices which are both unitary and Hermitian

Matrices such as $$ \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} \cos\theta & i\sin\theta \\ -i\sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} …
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Properties of zero-diagonal symmetric matrices

I'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices. The only thing I can come up with is that it cannot be positive definite (if it's not the zero matrix): The…
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Positive definite matrix must be Hermitian

Is there a simple way to show that a positive definite matrix must be Hermitian? I feel there is a long drawn out proof of this to be had by taking unit vectors and applying the positive definiteness property, and brute forcing it. But is there some…
pad
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Prove that Hermitian matrices are diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are orthogonal. If $A=A^H;\;\;\lambda_1,\lambda_2$ be…
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Can a symmetric matrix become non-symmetric by changing the basis?

We know that a hermitian matrix is a matrix which satisfies $A=A^*$, where $A^*$ is the conjugate transpose. A symmetric matrix (special case of hermitian - with real entries) is one for which $A=A^T$. Observation: this property is dependent on…
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A normal matrix with real eigenvalues is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not complete a proof. Could anybody please provide…
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How to prove this inequality for determinant of Hermitian block matrix?

I am given an Hermitian positive definite matrix $$D=\left(\begin{matrix}A&\overline{C}^T\\C&B\end{matrix}\right)$$ $A$ and $B$ are square matrices. The task is to prove the following…
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$A^2=A^*A$. Why is matrix $A$ Hermitian?

Let $A$ be $n \times n$ matrix and $A^2=A^*A$. Why is $A$ a Hermitian matrix?
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Eigenvalues of certain block Hermitian matrix

Suppose I have a special block, Hermitian matrix $$H = \begin{bmatrix} A & B \\ B^* & A^* \end{bmatrix}$$ where $*$ denotes conjugate transpose. The blocks $A$ and $B$ are themself Hermitian in this case. Are there any theorems considering the…
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What is a basis for the space of $n\times n$ Hermitian matrices?

So I was working on a specific problem related to Hermitian matrices. If we let $H_n$ denote the set of n x n Hermitian matrices. We're told that $H_n$ is a real vector space under matrix addition and scalar multiplication by a real number. I don't…
John Doe
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Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$ a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ I can define a companion matrix $$ A[p]=\begin{bmatrix} 0…
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Show $A$ is Hermitian and find the orthonormal basis for $V$ in which $A$ is diagonalizable.

Let $\{e_1,e_2,e_4\}$ be an orthonormal basis for a complex unitary space $V$. Let's define the vectors: $f_j=e_j-\frac14\sum\limits_{i=1}^4e_i, j\in\{1,2,3,4\}$. Let $A\in\mathcal L(V), Ax:=\sum\limits_{j=1}^4\langle x,f_j\rangle f_j$. Show $A$…
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What is dimension over $\mathbb R$ of the space of $n\times n$ Hermitian matrices?

what is dimension over $\mathbb{R}$ of $H_n( \mathbb{C})$, the set of $n \times n$ Hermitian matrices? My attempt: every real number is a complex number as all symmetric matrices are Hermitian. In my view the dimension of $H_n( \mathbb{C})$ is…
jasmine
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