# Are all Hermitian matrices normal?

The

**normal matrices**are the**matrices**which are unitarily diagonalizable, i.e., is a**normal matrix**iff there exists a unitary**matrix**such that is a diagonal**matrix**.**All Hermitian matrices**are**normal**but have real eigenvalues, whereas a general**normal matrix**has no such restriction on its eigenvalues.A.

### What is hermitian matrix and skew Hermitian matrix?

**Skew**-

**Hermitian Matrix**. A square

**matrix**, A , is

**skew**-

**Hermitian**if it is equal to the negation of its complex conjugate transpose, A = -A' . In terms of the

**matrix**elements, this means that. a i , j = − a ¯ j , i . The entries on the diagonal of a

**skew**-

**Hermitian matrix**are always pure imaginary or zero.

#### When a matrix is positive definite?

A positive definite matrix is a symmetric matrix with all positive eigenvalues.**Note**that as it's a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Now, it's not always easy to tell if a matrix is positive definite.#### What is a skew symmetric matrix?

In mathematics, particularly in linear algebra, a**skew**-**symmetric**(or**antisymmetric**or antimetric)**matrix**is a square**matrix**whose transpose equals its negative; that is, it satisfies the condition A^{T}= −A.#### What is meant by Nilpotent Matrix?

**Nilpotent Matrix**. There are two equivalent definitions for a**nilpotent matrix**. A square**matrix**whose eigenvalues are all 0. 2. A square**matrix**such that is the zero**matrix**for some positive integer**matrix**power , known as the index (Ayres 1962, p. 11).

B.

### Is a Hermitian matrix symmetric?

As a result of this definition, the diagonal elements of a

**Hermitian matrix**are real numbers (since ), while other elements may be complex.**Hermitian matrices**have real eigenvalues whose eigenvectors form a unitary basis. For real**matrices**,**Hermitian**is the same as**symmetric**.#### What is the use of identity Matrix?

The Multiplicative**Identity**. The**identity**property of multiplication states that when 1 is multiplied by any real number, the number**does**not change; that is, any number times 1 is equal to itself. The number "1" is called the multiplicative**identity**for real numbers.#### Can we find inverse of Nonsquare Matrix?

Invertible**matrix**.**Non-square matrices**(m-by-n**matrices**for which m ≠ n)**do**not have an**inverse**. However, in some cases such a**matrix**may have a left**inverse**or right**inverse**. If A is m-by-n and the rank of A is equal to n, then A has a left**inverse**: an n-by-m**matrix**B such that BA =**I**_{n}.#### What is the meaning of the determinant of a matrix?

A geometric interpretation can be given to the value of the**determinant**of a square**matrix**with real entries: the absolute value of the**determinant**gives the scale factor by which area or volume (or a higher-dimensional analogue) is multiplied under the associated linear transformation, while its sign indicates whether

C.

### Can a matrix be Hermitian and unitary?

Normal,

**Hermitian, and unitary matrices**. A square**matrix**is a**Hermitian matrix**if it is equal to its complex conjugate transpose . If a**Hermitian matrix**is real, it is a symmetric**matrix**, . is a**unitary matrix**if its conjugate transpose is equal to its inverse , i.e., .#### What is the conjugate transpose of a matrix?

In mathematics, the**conjugate transpose**or**Hermitian transpose**of an m-by-n matrix A with complex entries is the n-by-m matrix A^{∗}obtained from A by taking the**transpose**and then taking the complex**conjugate**of each entry. (#### What is a unitary operator?

A**unitary operator**is a bounded linear**operator**U : H → H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H → H is the identity**operator**. Thus a**unitary operator**is a bounded linear**operator**which is both an isometry and a coisometry, or, equivalently, a surjective isometry.#### What is a self adjoint operator?

In mathematics, a**self**-**adjoint operator**on a finite-dimensional complex vector space V with inner product is a linear map A (from V to itself) that is its own**adjoint**: . Since an everywhere-defined**self**-**adjoint operator**is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case.

Updated: 7th December 2019