as much as this point, we have advanced in our examine of linear algebra without ever before specifying even if it is the entries of our vectors and also matrices are real or facility numbers. Back the examples and also exercises presented therefore far concern real matrices (i.e., matrices having actually real entries), all the definitions, propositions and results discovered in vault lectures are applicable without modification to facility matrices (i.e., matrices whose entries are complex numbers). In fact, if you revise those lectures, you will realize that nowhere, and also especially in no proof, that is crucial to assume that a procession or vector be real. The only caveat is that when we deal with complicated matrices, we additionally need to use complicated scalars when taking direct combinations.

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In this lecture, we space going come revise some elementary facts about facility numbers. We then show some basic properties of facility matrices and carry out some helpful definitions.


Table that contents

Complex numbers

Complex matrices

Solved exercises

Complex numbers

A facility number is a number that have the right to be created as" style="background-position:0px -1320px;"/>where and are genuine numbers, called the real and imaginary component of the complicated number respectively, and" style="background-position:0px -1434px;"/>is called imaginary unit.

Hence, as soon as we manipulate complicated numbers, the crucial "trick" we exploit over and over again is " style="background-position:0px -1512px;"/>

imaginary numbers permit to discover solutions to equations that have no genuine solutions. Because that example, the equation" style="background-position:0px -1474px;"/>has no genuine solution, but it has actually two imagine solutions" style="background-position:0px -1650px;"/>

real numbers are complicated numbers that have actually zero imagine part. The latter is regularly omitted, that is, rather of creating we just write .

Complex conjugate

vital concept is the of complex conjugate. Provided a complicated number" style="background-position:0px -1320px;"/>its conjugate, denoted through , is" style="background-position:0px -1358px;"/>

as a consequence, twin conjugation pipeline numbers unchanged:" style="background-position:0px -135px;"/>

Algebra of complicated numbers

The algebra of facility numbers is similar to the algebra of actual numbers. Given two complicated numbers" style="background-position:0px -986px;"/>we have actually the following rules:


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Distributive properties of conjugation

keep in mind that shortcut is distributive under addition:

" />and under multiplication:
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Modulus of a facility number

The modulus (or absolute value) of a complex number is identified as" style="background-position:0px -1px;"/>

where we take into consideration only the positive root.

Clearly, the modulus is constantly a genuine number.

Complex matrices

facility matrices (and vectors) space matrices whose entries are facility numbers.

Complex shortcut of matrices

offered a procession , its facility conjugate is the matrix such the " style="background-position:0px -1396px;"/>that is, the -th entry of is equal to the complicated conjugate of the -th entry of , for any type of and .

Example define the procession " style="background-position:0px -221px;"/>Then its complicated conjugate is" style="background-position:0px -576px;"/>

Distributive nature of conjugation

The distributive nature that hold for the conjugation of facility numbers hold also for the shortcut of matrices.

Proposition If and also space two matrices, then" style="background-position:0px -945px;"/>


We have actually that " style="background-position:0px -1050px;"/>for any type of and also , by the distributive residential or commercial property of the link of facility numbers under addition.

Proposition If is matrix and also is a matrix, then" style="background-position:0px -1185px;"/>

We have that " style="background-position:0px -727px;"/>for any kind of and , by the distributive building of the conjugation of complicated numbers under addition and multiplication.

Proposition If is a matrix and is a scalar, then" style="background-position:0px -1144px;"/>

We have" style="background-position:0px -1226px;"/>for any type of and , by the definition of multiplication that a matrix by a scalar and also by the distributive property of the link of complex numbers under multiplication.

Conjugate that a actual matrix

A trivial but beneficial property is that taking the conjugate the a matrix that has only actual entries walk not adjust the matrix. In other words, if has actually only genuine entries, then" style="background-position:0px -1612px;"/>

This is a repercussion of the truth that a real number have the right to be seen as a complex number through zero imaginary part. Yet all the conjugation go is to readjust the sign of the imaginary component of a complicated number. Therefore, a genuine number is equal to the conjugate.

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Solved exercises

below you can uncover some exercises with explained solutions.

Exercise 1

specify two vectors

" />Compute the complying with modulus: " style="background-position:0px -1750px;"/>


The product of and also is

" />and that modulus is" style="background-position:0px -175px;"/>

Exercise 2

define " style="background-position:0px -1574px;"/>and" style="background-position:0px -653px;"/>Compute " style="background-position:0px -1712px;"/>