Stepper motor

In mathematics, a complex square matrix A is normal if

$A^{*}A=A{A}^{*}$

where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.

A matrix A with real entries satisfies A*=A^T, and is therefore normal if A^TA = AA^T.

Normality is a convenient test for diagonalizability: a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A=AA* is diagonalizable.

The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.

Special cases

Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal.

However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian. As an example, the matrix

$A=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right)$

is normal because

$A{A}^{*}=\left(\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ 1& 1& 2\end{array}\right)=A^{*}A.$

The matrix A is neither unitary, Hermitian, nor skew-Hermitian.

The sum or product of two normal matrices is not necessarily normal. If they commute, however, then this is true.

If A is both a triangular matrix and a normal matrix, then A is diagonal. This can be seen by looking at the diagonal entries of A^*A and AA^*, where A is a normal, triangular matrix. Say A is upper-triangular. Because (A^*A)_ii=(AA^*)_ii, the first row must have the same norm as the first column,

$\left|\right|A{e}_1|{|}^2=|\left|A^{*}{e}_1\right|{|}^2$.

The first entry of row 1 and column 1 are the same, and the column 1 is zero for entries 2 through n. This implies the first row must be zero for entries 2 through n. Continuing this argument for row column pairs 2 through n shows A is diagonal.

Consequences

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies: a matrix A is normal if and only if it can be represented by a diagonal matrix Λ and a unitary matrix U by the formula

$\mathbf{A}=\mathbf{U}\mathrm{\Lambda }{\mathbf{U}}^{*}$

where

$\mathrm{\Lambda }=\mathrm{diag}({\lambda }_1,{\lambda }_2,\dots )$

${\mathbf{U}}^{*}\mathbf{U}=\mathbf{U}{\mathbf{U}}^{*}=\mathbf{I}.$

The entries λ of the diagonal matrix Λ are the eigenvalues of A, and the columns of U are the eigenvectors of A. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of U.

Another way of stating the spectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of Cⁿ. Phrased differently: a matrix is normal if and only if its eigenspaces span Cⁿ and are pairwise orthogonal with respect to the standard inner product of Cⁿ.

The spectral theorem for normal matrices can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition. In fact, let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B. But then B must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal.

The spectral theorem permits the classification of normal matrices in terms of their spectra. For example, a normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane. Also, a normal matrix is self-adjoint if and only if its spectrum consists of reals.

In general, the sum or product of two normal matrices need not be normal. However, there is a special case: if A and B are normal with AB = BA, then both AB and A + B are also normal. Furthermore the two are simultaneously diagonalizable, that is: both A and B are made diagonal by the same unitary matrix U. Both UAU^* and UBU^* are diagonal matrices. In this special case, the columns of U^* are eigenvectors of both A and B and form an orthonormal basis in Cⁿ. This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.

Equivalent definitions

It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let A be a n-by-n complex matrix. Then the following are equivalent:

1. A is normal.
2. A is diagonalizable by a unitary matrix.
3. The entire space is spanned by some orthonormal set of eigenvectors of A. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
4. $\Vert Ax\Vert =\Vert A^{*}x\Vert$ for every x.
5. $\mathrm{tr}(A^{*}A)=\sum _j^n|{\lambda }_j{|}^2.$ (That is, the Frobenius norm of A can be computed by the eigenvalues of A.)
6. The Hermitian part $\frac{1}{2}(A+A^{*})$ and skew-Hermitian part $\frac{1}{2}(A-A^{*})$ of A commute.
7. $A^{*}$ is a polynomial (of degree ≤ n − 1) in $A$.
8. $A^{*}=AU$ for some unitary matrix U.
9. U and P commute, where we have the polar decomposition A = UP with a unitary matrix U and some positive semidefinite matrix P.
10. A commutes with some normal matrix N with distinct eigenvalues.
11. ${\sigma }_i(A)=|{\lambda }_i(A)|$ for all $i=1...n$ where A has singular values ${\sigma }_1(A)\ge ...\ge {\sigma }_n(A)$ and eigenvalues $|{\lambda }_1(A)|\ge ...\ge |{\lambda }_n(A)|$

Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is only quasinormal.

The operator norm of a normal matrix N equals the numerical and spectral radii of N. (This fact generalizes to normal operators.) Explicitly, this means:

${\sup }_{\Vert x\Vert =1}\Vert Nx\Vert ={\sup }_{\Vert x\Vert =1}|⟨Nx,x⟩|=\max \{|\lambda |:\lambda \in \sigma (N)\}$

Analogy

It is occasionally useful (but sometimes misleading) to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:

• Invertible matrices are analogous to non-zero complex numbers
• The conjugate transpose is analogous to the complex conjugate
• Unitary matrices are analogous to complex numbers whose absolute value is 1
• Hermitian matrices are analogous to real numbers
• Hermitian positive definite matrices are analogous to positive real numbers
• Skew Hermitian matrices are analogous to purely imaginary numbers

(As a special case, the complex numbers may be embedded in the normal $2\times 2$ real matrices by the mapping $a+bi\mapsto \left(\begin{array}{cc}a& b\\ -b& a\end{array}\right)$, which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.)

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.