# What is Matricies: Definition and 40 Discussions

This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to diverse ways of classifying matrices. A first group is matrices satisfying concrete conditions of the entries, including constant matrices. Important examples include the identity matrix given by

I

n

=

[

1

0

0

0

1

0

0

0

1

]

.

{\displaystyle I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.}
and the zero matrix of dimension

m
×
n

{\displaystyle m\times n}
. For example:

O

2
×
3

=

(

0

0

0

0

0

0

)

{\displaystyle O_{2\times 3}={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}}
.Further ways of classifying matrices are according to their eigenvalues, or by imposing conditions on the product of the matrix with other matrices. Finally, many domains, both in mathematics and other sciences including physics and chemistry, have particular matrices that are applied chiefly in these areas.

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1. ### Prove the identity matrix is unique

I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...
2. ### Linear Transformations if the design matrix

Homework Statement given that X is an n × p matrix with linearly independent columns. And $$X^∗ = XA$$ where A is an invertible p × p matrix. a) Show that: $$X^*{({X^*}^TX^*)^-}^1{X^*}^T = X{(X^TX)^-}^1X^T$$ b) Consider two alternative models $$M : Y = Xβ + ε$$ and M^∗ : Y = X^∗β ^∗ +...
3. ### B ##AB = I \implies BA = I##, for square matricies ##A,B##

Let ##(AB)_j## be the jth column of ##AB##, then ##\displaystyle (AB)_j = \sum^n_{r= 1} B_{rj} \alpha_r## where ##\alpha_r## is the rth column of ##A##. Also ##(BA)_j = B \alpha_j \implies A(BA)_j = \alpha_j## susbtituting this in the sum ##\displaystyle (AB)_j = \sum^n_{r = 1} B_{rj}A(BA)_r##...
4. ### Matrix Operations: Inverse Existence & Row Op.

Homework Statement [/B] \begin{array}{cc}1 & 1&1\\ 1&1-s&1-s\\-s&1-s&s^2-1\end{array} a)For which values of s does the inverse exist, and why? You should be using row operations and ideally head for reduced row echelon form b) In the process of calculating part a), you will come across a...
5. ### System of Equations Homework: 4 Unknowns, 5 Equations

Homework Statement Okay this is the problem it seems so easy but i just cannot for the life of me get it to click into my mind, I have 4 unknowns and 5 equations and i have to put it into a matrix and try solve it matricies or eigenvalues/eigenvectors. The 5 equations are: a= b/2 b=a/3 + d...
6. ### Can quaternion group be represented by 3x3 matricies?

Hi, The Quaternion group, ##Q=\{1,-1,i,-i,j,-j,k,-k\}##, can be realized by ##2x2## matricies: ## \begin{align*} 1=\begin{bmatrix} 1,0 \\ 0,1\end{bmatrix} &\hspace{10pt} i=\begin{bmatrix} \omega,0 \\ 0,-\omega\end{bmatrix} & \hspace{10pt}j=\begin{bmatrix} 0,1 \\ -1,0\end{bmatrix} &...
7. ### MATLAB MATLAB trouble with reshape and random matricies

I am trying to learn MATLAB with MIT OCW and I am running into some trouble. It says as an assignment: c. cMat = a 10x10 matrix where the vector 1:100 runs down the columns (use reshape). so 1 11 21...91 2....92 . ..... 10...100 is the matrix I am trying to make and another...
8. ### Looking for Matricies with their R-Echelon Forms

Hi all, I'm testing out a matrix solving program and while it checks out for 2x2/3x3/4x4 I would like to try it out on some larger matrices, but I don't really want to go through the hassle of row reducing a couple of 10x10 matrices to double check my program. Does anyone happen to know of...
9. ### What Makes a Hermitian Matrix Positive Definite?

on page 261 of this paper by J. Vermeer (http://www.math.technion.ac.il/iic/e..._pp258-283.pdf ) he writes The following assertions are equivalent. a) A is similar to a Hermitian matrix b) A is similar to a Hermitian matrix via a Hermitian, positive definite matrix c) A is similar to A*...
10. ### Can I prove matrix properties using simple steps?

Homework Statement Homework Equations The Attempt at a Solution -(y, x) = -(YX-XY) = XY-YX Can I do this or would I have to define a matrix X= ( a b c d ) Y= ( e f g h)...
11. ### Engineering Mesh Current circuit calculations with matricies

Homework Statement Using matrix algebra calculate the voltage expected across each resister in the circuit diagram. Only need the simultaneous equations. The Attempt at a Solution -15=6800(I_b-I_a )+2200(I_b-I_c) 15=2200(I_c-I_b )+8200(I_c-I_d) -25=8200(I_d-I_c )+4700I_d Thanks for any help.
12. ### Quantum Computing gate matricies

Hi there I am working through a quantum gate section of my course and I am a bit puzzled on how to calculate a matrix for consecutive quantum gates. I understand how to generate a matrix for |q0⟩--------[H]------- |q1⟩------------------ Which is simply the tensor product of the hadamard...
13. ### Matrix Multiplication and Evaluation: B & A

where possible evaluate the following B A B= (12 14 15 ) A= ( 1.2 1 ) ( 1 1 12 ) (-0.6 0.8 ) ( -0.1 -0.9 )
14. ### MATLAB How to Create a Multiplication Matrix in MATLAB?

Hi, I'm new to MATLAB and I have a problem: I am trying to create a multiplication matrix of sorts. The first row will have numbers from 0 to 1 in steps of .0101 The first column will have the same, from 0 to 1 in steps of .0101 Then to generate the rest I will multiple each entry in the...
15. ### Orthogonal Matricies and rotations.

Hi all, I have been trying to gain a deeper insight into quadratic forms and have realized that my textbook makes the assumption that an orthogonal matrix corresponds to either a rotation and/or reflection when viewed as a linear transformation. The textbook outlines a proof that demonstrates...
16. ### Expressing matricies as vectors

G'day, I'm doing some questions on whether or not a given set spans a given vector space and was wondering what the best way to write out a matrix is. For example, if I am wanting to show that a bunch of M2X2 elements spans M2X2, can I express each matrix as a coordinate vector with respect to...
17. ### Find P Such That P^-1AP=B: Similar Matricies

1a) First find if A and B are similar (ie: A~B). b) If so find P such that P(^-1)AP=B. (P^-1 is the inverse of P) Ok so I'm not going to give the matricies because I don't know how to write them out properly on this and It doesn't really matter anyways. First I found if A and B were...
18. ### How does one calculate the Tensor product of two matricies?

Just as a concrete example, say A and A' are two 2x2 matricies from R^2 to R^2, A = \left [ \begin{array}{cc} a \,\, b \\ c \,\, d \end{array} \right ] A' = \left [ \begin{array}{cc} x \,\, y \\ z \,\, w \end{array} \right ] What would A \otimes_\mathbb{R} A' look like (say wrt the standard...
19. ### Higher Dimensional Dirac Matricies

Homework Statement If D =7 and the metric g\mu\nu=diag(+------), Using the outer product of matrices, A \otimes B construct a suitable set of \gamma matrices from the 2 x 2 \sigma-matrices Homework Equations \sigma1=(0, 1 ) \sigma2=(0, -i)...
20. ### Linear algebra identities of inverse matricies

Homework Statement Left Inversion in Rectangular Cases. Let A^{-1}_{left} = (A^{T}A)^{-1}A^{T} show A^{-1}_{left}A = I. This matrix is called the left-inverse of A and it can be shown that if A \in R^{m x n} such that A has a pivot in every column then the left inverse exists. Right...
21. ### Prove Existence of Real Invertible Matrix Q for A & B 2x2 Similar Matricies

Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP. Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ. A and B are similar when thought of as complex matricies, so they represent the same...
22. ### Solving Similar Matricies: Find Real Invertible 2x2 Matrix Q

Homework Statement Let A and B be 2x2 real matricies, and suppose there exists an invertible complex 2x2 matrix P such that B = [P^(-1)]AP. Show that there exists a real invertible 2x2 matrix Q such that B = [Q^(-1)]AQ. Homework Equations A and B are similar when thought of as...
23. ### Exploring Dirac Matrices in the Context of the Dirac Equation

I wonder if I can chose any 4x4 matrices \gamma^\mu which fullfil anticommutationn relations \{\gamma^\mu,\gamma^\nu \}=2g^{\mu\nu} as a matricies in Dirac equation: i \gamma^\mu \partial_\mu \psi= m \psi . What changes in the theory if I chose different matricies? (of course I have to...
24. ### How many 2x2 matricies equal I?

Assuming A is a 2x2 matrix how many different matricies exist such that A^2=I ? I am 99% sure the answer is 4 but after putting that down as an answer with supporting evidence I was marked wrong (or atleast not fully correct) so I am stumped as to where to jump and whether or not the grader may...
25. ### Dirac Gamma Matricies and Angular Momentum Commutation Relations

Homework Statement This isn't really the problem, but figuring this out will probably help me with the rest of the problem. I want to know what [\gamma^0, L_x] is. Homework Equations I know the commutation (or rather anticommutation) relations between the gamma matricies, and I know the...
26. ### Rotation Matrix: Calculating Angle & Direction of Rotation

Homework Statement Hey guys, I'm not sure if this bit is relevant but the first part of the question is... 'The diagram shows a triangle with vertices O, A(1,2) , B (0,2). The question I need help with is 'Each of the following matricies represents a rotation about the origin, Find the angle...
27. ### Eigenvalues and Eigenvectors of 3x3 matricies

Hello Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues. Can someone please help.. Thanks
28. ### MATLAB MATLAB: Average a large number of matricies from .mat files

I have a series of large 2x2 matricies, each of which is stored inside a .mat file. These files have the names data1.mat, data2.mat, data3.mat,..., data60.mat. I have sucessfully loaded each of these .mat files. I want to create a 1x60 array whose entires are the average values of the...
29. ### Triangular matricies and subspaces

hello again I was asked if the set of all uppertriangular nxn matricies are a subspace of Mnn, how would you check if it has a zero vector and closed under addition and multiplication ? and why did they ask for the upper triangular matrix instead of the lower one? or either
30. ### Express this 2x2 matrix as a linear combo of thoes other 2x2 matricies

http://xmlearning.maths.ed.ac.uk/lecture_notes/vector_spaces/linear_span_set/linear_span_set.php Problem 1.29 #3------------------------ Hello guys, I've been given a problem like the one above in my assignments and I'm not really sure what the final answer is supposed to be like. So far here...
31. ### Help with Wronskians and matricies?

http://www.math.ucdavis.edu/~lai/22b/handouts/Wronskian.pdf *wrong link I am working off this example. First off I think there is a typo in it, in the second example y1'' should be two, can someone confirm this because at the moment i am losing my mind? My question is how or...
32. ### C/C++ Solving Hermitian matricies in C/C++

I've been struggling for awhile, I've been trying to use CLAPACK to avoid learning Fortan. I think I've just a linking problem, since I've been testing code that's supposed to work. in the VC command prompt i type cl dgesv.c and I get the error LNK2019: unresolved external symbol...
33. ### Prove/Disprove: Similar Matricies w/ Zero Rows

Homework Statement Prove or disprove the following statement: If A is a singular matrix (detA=0) the it's similar to a matrix with a row of zeros. Homework Equations The Attempt at a Solution I know that A has an e-value 0 which means that it's similar to a matrix that has a...
34. ### Complex numbers, matricies and Kirchoff's laws

Hello everyone.. I have quite a problem regarding A.C. circuit analysis using complex numbers and 2x2 matricies. * The aim is to find the current in each of the two loops and apply Kirchoff's laws. I believe the overall aim is just to prove that the laws are actually in place.. (SEE...
35. ### Finding Variables within Matricies Question

:confused: Hi There! I have a quick question and I would really appreciate some guidence! I have three matrix multiplied together like so (horizontal matrix 1 k)(2x2 matrix 3 4 -2 1)(vertical matrix 1 k) equals 11 I'm asked to find two values for k. I've tried multiplying them out...
36. ### Applications of singular matricies

Are there any applications that always involve the use of noninvertible or singular matrices?? I know there are plenty for invertible ones. Thanks.
37. ### Sudoku solving with matricies and/or diophantine equations

I was wondering if there is a way to solve these puzzles with matricies and/or diophantine equations: http://www.sudoku.com/ If you define the basis as nine orthogonal vectors, and input the given initial values to the corresponding places in a 9X9 matrix, and also brake it up into 9 3X3...
38. ### What Are the Various Applications of Matrices Beyond Solving Equations?

I've only taken one linear algebra course, and when I was tought about matricies, I immediatly wondered what the point of them was. I understand they can be useful for solving systems of equations, but there must be other things they can be used for. Probably many. ...what are these things?
39. ### Is ABCDEF = AB(C(D))EF for matricies

is ABCDEF = AB(C(D))EF for matricies? Also is ABCD(EFGH) = ABCDEFGH? Thanks in advance!