For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

learn more… | top users | synonyms (2)

2
votes
1answer
31 views

How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
0
votes
1answer
32 views

If a NxN matrix has two identical columns will its determinant be zero?

I am currently doing a practice final for a Linear Algebra Class. In it I am given the following statement and asked to determine whether it is true or false. "If det(A) = 0, then two rows or two ...
0
votes
0answers
21 views

Manipulating product of two matrices

In a published paper I saw the following $$\log \left(\mathbf{I} + \mathbf{T}\mathbf {Hpp^HH^H}\right)= \log(1+\mathbf {p^HH^HTHp})$$ where uppercase means a matrix while lower case means vector ...
-5
votes
1answer
30 views

finding the product elementary matrices. [closed]

consider the matrix $$ Factor the matrices as a product of elementary matrices. $$ \begin{pmatrix} 2 & 0 & 0\\ 0 & 3 & -1\\ 10 & 12 & 3\\ ...
0
votes
2answers
39 views

Decomposition of an unitary operator by simple operators

For quantum computation, it's well known that any unitary operator can be approximated with an arbitrary accuracy by simple operators, for example to approximate an unitary operator on n qubits by no ...
0
votes
3answers
41 views

Reducing to upper triangular form

I've just had some difficulty with this transforming this matrix into upper triangular form: $$ \pmatrix{ i& 2i& -1\\1 & 1& i\\ 2-i& 1& i } $$ I've tried almost everything. ...
1
vote
0answers
27 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...
1
vote
0answers
21 views

Derivative of a determinant with respect to a matrix

Can someone tell me the derivative of the following determinant ($\Psi\in\mathbb{R}^{p\times p}$, $Z\in\mathbb{R}^{p\times q}$, $\alpha\in\mathbb{R}^q$) $\frac{\partial}{\partial \Psi} ...
1
vote
1answer
59 views

About the Jordan Form

So i have a few questions about the Jordan form. Say we have a matrix $A$ and has λ1 λ2..λκ eigenvalues.Why is it Usefull to know the index of the matrices $A-λιI$ ? Also i have seen jordan forms ...
-1
votes
1answer
20 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ ...
1
vote
2answers
29 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
5
votes
2answers
65 views

Find trace of linear operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$ by permutation of the basis vectors. Suppose we know its eigenvalues ( some roots of unity ): ...
1
vote
2answers
35 views

Why does positive definite matrix have strictly positive eigenvalue?

We say $A$ is a positive definite matrix if and only if $x^T A x > 0$ for all nonzero vectors $x$. Then why does every positive definite matrix have strictly positive eigenvalues?
-3
votes
0answers
107 views

Let A be an n×m matrix and B an m×k matrix. Show that AB = 0 if and only if Col(B) ⊆ Nul(A). [closed]

Need help asap, any help would be appreciated. Do not even know how to start. full answer preferred.
1
vote
3answers
48 views

How is that a rotation by an angle θ about the origin can be represented by this transformation matrix?

$$ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} $$ How was this matrix derived? I know how to use it, but where did it come from? Can someone prove why ...
-1
votes
1answer
31 views

Inner product and matrix properties [closed]

For vectors $v_1,...,v_n$ in $R^n$, how to show that $$\sum_{j=1}^n\langle u,v_j\rangle v_j=(v_1 \ v_2 \ ... \ v_n)(v_1^T \ ... \ v_n^T)u$$ I got hang up on the calculation of inner product. :( Any ...
2
votes
2answers
55 views

Why does $\frac{\partial a^TX a}{\partial X} = aa^T$?

$$\frac{\partial a^TX a}{\partial X} = \frac{\partial a^TX^T a}{\partial X} = aa^T \tag 1$$ I got (1) from the Matrix Cookbook. But I don't see how you derive it? Why isn't it $a^Ta$. Assume that ...
0
votes
1answer
39 views

covariance matrix of X+Y and X-Y

This question comes up in almost every past paper i do and is worth 10 marks and just can't work it out... Let $X$ and $Y$ have the joint pdf $$f(x,y)= \begin{cases} e^{-y}, \text{if} \ 0 < x ...
2
votes
1answer
50 views

Which power of an integer matrix is identity modulo $p^\alpha$?

I've read this question about identity power of an integer matrix. But how about power of a matrix modulo $p^\alpha$. $$A^m \equiv I \pmod{p^\alpha} $$ How can I find the minimal $m$ that the above ...
-4
votes
4answers
89 views

Determinants with arithmetic progressions as columns [closed]

Prove that determinants of the following form all vanish: $$\det \begin{bmatrix} x-3 & x-4 & x-a \\ x-2 & x-3 & x-b \\ x-1 & x-2 & x-c\end{bmatrix} = 0$$ Here $a$, $b$, $c$ are ...
-6
votes
0answers
37 views

I hope you can solve this question because I really have no idea what the question is all about [closed]

Prove that matrices |x-3 x-4 x-a| which is first row,|x-2 x-3 x-b|which is second row and |x-1 x-2 x-c| which is third row = 0 where a b and c are three consecutive terms in arithmetric progression
0
votes
0answers
43 views

Jordan Form of generic matrix

Say $ A\in\mathbb{C}^{6\times6} $ and has eigenvalues $\lambda_1$ and $\lambda_2$ of multiplicity $ 3$ both of them. And for $\kappa=1,2,3$ the echelon form of the matrix $$ (A-\lambda_1I)^\kappa $$ ...
-1
votes
1answer
22 views

$D\det_A$ exists and equals $D\det_A (H)=\det (A) \operatorname{trace} (A^{-1}H) $? [closed]

Consider the determinant function $\det : M_n(\mathbb R ) \to \mathbb R$ , the is it true that $D\det _{A}$ exists ? Does it exist if $A$ is assumed to be invertible also and at $H \in M_n(\mathbb R)$ ...
8
votes
1answer
147 views

Prove that, if $A, B$ are matrices from $M_4(R)$ so that $AB=BA$

Prove that, if $A, B$ are matrices from $M_4(\Bbb R)$ so that $AB=BA$ and $\det(A^2 −AB + B^2) = 0$ then: $$ \det(A + B) + 3\det(A − B) = 6 (\det(A) + \det(B)) \tag 1 $$ What I tried: Because of ...
0
votes
0answers
18 views

Using non-standard inner products for alternative notions of matrix product

It seems intuitive to think of billinear forms on finite dimensional vector spaces as coresponding to positive definite, symmetric or hermitian matrices. In this language, the standard inner product ...
1
vote
0answers
35 views

How to realise $\mathrm{PGL}_2$ as a closed subgroup of some $\mathrm{GL}_n$ explicitly?

Let $k$ be an algebraically closed field, then it is well-known that any affine algebraic group $G$ over $k$ can be viewed as a closed subgroup of $\mathrm{GL}_n$ for some $n$. In the special case ...
1
vote
0answers
14 views

Permutation unitary in a tensor product

Given a matrix of the form $$ A = B_{1} \otimes B_{2} \otimes B_{3} \otimes... \otimes B_{n} $$ how can I find a matrix that gives me a permutation of , say, two of the elements: $$ A = B_{2} ...
1
vote
1answer
37 views

Prove idempotent and invertible

A square matrix $A$ is idempotent if $A^2 = A.$ Prove that if $A$ is an $n\times n$ matrix that is idempotent and invertible, then $A$ is the identity. How do i prove this?
0
votes
0answers
15 views

Spectrum of the circulant graph

How to prove that the eigenvalue of cycle $C_n=\lambda_r=2 cos(2\pi r/n)$?where $r=0,1,...n-1$, which is proved for the circulant matrix with first row $(v_0=0,v_1=1,v_2=0, ...v_{n-2}=0,v_{n-1}=1)$, ...
0
votes
1answer
15 views

Finding unkonwn values in a matrices multiplication.

I have an unknown 3x3 matrix multiplied by a known 3x1 matrix. I also know the resulting matrix . How would you go by solving for the values, or possible values, in the unknown matrix. $\left( ...
0
votes
1answer
19 views

Is there a name for a general upper triangular hollow matrix?

A hollow matrix is one with zero diagonal elements (according to this web page) Q1: Is there a name for an upper (or lower) triangular hollow matrix? Q2: Alternatively how might such an object be ...
1
vote
2answers
31 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
0
votes
1answer
43 views

In $P_2 = {ax^2 + bx + c: a,b,c \in\mathbb{R}}$, why do coefficients in $ax^2$ form reduce with coefficients in $bx$ or $c$?

In $P_2 = {ax^2 + bx + c: a,b,c \in\mathbb{R}}$, why do coefficients in $ax^2$ form reduce with coefficients in $bx$ or $c$? For example, lets look at the set {$x^2 + x - 1, 2x + 1, 2x - 1$} If we ...
1
vote
1answer
57 views

Solving for multiple variables given a couple of equations?

I have a panel 1200 pixels wide, and am filling in smaller subpanels to fill the length. Each sub-panel is of a different color ($p$ = purple, $g$ = green, etc). It's for a navigation bar on a ...
1
vote
0answers
23 views

Using the power iteration method to compute the largest eigen value for hermitian matrix

I have read a published paper that does the following: Starts with $\bf G$ being a Hermitian matrix The authors then used the power iteration over G to find the largest eigen value as follows: ...
-1
votes
1answer
23 views

a question how to compute the eigenvalues of a matrix [duplicate]

I have a question: Suppose I have a $n\times n$ matrix: $$ \begin{bmatrix} 1 & 1 &...& 1 \\ 1 & 1 &...&1 \\ \vdots&\vdots &\ddots & ...
2
votes
2answers
32 views

Generate a semi-unitary matrix

I would like to generate a semi-unitary matrix, i.e., $UU^T=~I$ where U is a non-square matrix whose number of rows is bigger than its number of columns. I tried it by solving the optimization problem ...
2
votes
4answers
37 views

Decide if each is a basis for $P_2$. (a) $(x^2 + x - 1, 2x + 1, 2x - 1)$

I'm using Linear Algebra by Jim Hefferon (freely available, links below with solution). I'm having trouble understanding Exercise 1.18 on page 117. 1.18 Decide if each is a basis for $P_2$. (a) ...
0
votes
0answers
7 views

Cross-verifying a homography on known correspondences

Context I have two sets of known 2D correspondences $S_1$ and $S_2$, from which I have constructed homographies $h_1$ and $h_2$. This was achieved using the homogeneous estimation method, ie. by ...
7
votes
1answer
45 views

Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class ...
0
votes
2answers
44 views

Calculate matrix A from null space basis of $A-4I$

How to find a matrix $A$ when you are given some parameters and the basis for the null space? The problem I've been scratching my head over is this. The basis for the null space of $A-4I$ is ...
1
vote
1answer
50 views

Find eigenvalues of operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$. Suppose we know its eigenvalues - $\lambda_1, \lambda_2, \ldots, \lambda_n.$ Now consider the ...
4
votes
6answers
83 views

$A_{n\times n}$ that implies : $A^2-2A+I=0$ Proof $1$ is an eigevalue of $A$

I have the following question : Let $A_{n \times n}$ that implies : $A^2-2A+I=0$ Proof $1$ is an eigevalue of $A$ I don't really know how to approach this this what I manage to do (its not much ...
3
votes
1answer
60 views

proximal operator of weighted L1 norm

I hope to solve this problem. $$\min \quad \left\| CX \right\|_{1} $$ $$ \text{s.t.}\quad AX=b, X >0 $$ where $C \in \mathbb{R}^{m \times m}$, $X \in \mathbb{R}^{m \times n}$, $A \in ...
-2
votes
1answer
51 views

Proof: A matrix with $m$ rows and $n$ colums has $nm$ entries.

How to prove rigorously the following statement: A matrix (a collection of numbers $a_{ij}:1\leq i \leq m, 1\leq j \leq n)$ with $m$ rows and $n$ colums has $nm$ entries. By rigorously I mean ...
0
votes
0answers
26 views

Why is a positive definite matrix needed in the ellipsoid matrix representation?

An ellipsoid centered at the origin is defined by the solutions $\mathbf{x}$ to the equation $\mathbf{x}^TM\mathbf{x} = 1$, where M is a positive definite matrix. How can I see why M needs to be ...
-1
votes
0answers
21 views

Linear Algebra (Norm 2) [closed]

$$Q=2 ||B C^2 x|| $$ (2: is the norm two). How I can find Q in function of $B(m \times m),C(m \times m)$ and $x(m \times 1)$ (B and are matrices and x is a vector). I thank you in ...
1
vote
0answers
18 views

Can more pertubations in eigenvalues/vectors lead to smaller changes?

Say i have a $n$ x $n$ matrix $M$, and i change it's smallest eigenvalue from a small negative value $v$ to a small positive value $t$ to obtain $M^*$: $$M^* = VE^*V'$$ $E^*$ is a diagonal matrix of ...
0
votes
0answers
15 views

A class of sparse matrices whose inverse is also sparse?

In general the inverse of a sparse matrix is dense. A notable (but trivial) exception from that rule are diagonal matrices. Is there any other (broad) class of sparse matrices whose inverse is also ...
0
votes
1answer
17 views

Probability measure on the space of $n \times n$ symmetric matrices with non negative integer coefficients

I know that there exists a particular measure, called Haar measure, defined on random matrices, i.e. $n \times n$ orthogonal complex matrices. My question is the following: can we define a ...