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), ...

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22 views

Calculating the Span of a Matrix in MATLAB

If I have a matrix (or a set of vectors) say A=[1 2 4] [2 9 8] [7 9 3] how can I calculate its span in MATLAB? There is no direct command for it? Do I have to form a set of linear ...
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0answers
15 views

A question in perturbation of $P(\lambda )$

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
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0answers
12 views

Inner Product of Square Matrices

Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{n1} & \cdots ...
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0answers
14 views

Dot Product of Square Matrices & Inner Product

I need some help! Thank you in advance. Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots ...
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1answer
31 views

Tensors in matrix multiplication algorithms

Fast matrix multiplication algorithms, be it the Winograd and Coppersmith algorithm or any further improvement of it, extensively use tensors. In fact, the entire construction is based on tensor ...
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2answers
29 views

Find values of $t$ for which a matrix is invertible?

$$M = \begin{pmatrix} 2-t & 0 & 0 & 3\\ 0 & t & -t & 0\\ 0 & -t & 2t & 0\\ t-2 & 0 & 0 & t+3 \end{pmatrix}$$ Using upper triangular form ...
1
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3answers
69 views

Find the values of 'a' in a $4\times 4$ matrix(A) when the determinant is less than 2012

The matrix is $A \ =\begin{pmatrix} 7 & 1 & 3 & -2\\ -2 & 1 & -12 & -1 \\ 1 & 16 & -4 & a \\ ...
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1answer
33 views

Reduced-row echelon form associated to three lines in the plane

Let $\ell_1,\ell_2$ and $\ell_3$ be three lines in the plane $\mathbb{R}^2$. For $i = 1, 2, 3$, let the line $\ell_i$ have equation $a_i x + b_i y = c_i$. Is it possible for the matrix $$ ...
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1answer
41 views

How to find the symmetric matrix if its eigenvalues and eigenvectors are given?? [closed]

Find a $2 \times 2$ symmetric matrix if its eigenvalues are $1$ and $3$ and its corresponding eigenvectors are $(1,-1)$ and $(1,1)$.
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1answer
42 views

Sum of powers of a matrix with primitive polynomial modulo $2^{r}$

I need to prove an statement in the matrix form, which leads to the following equality modulo $2^{r}$. Which I couldn't prove but with computer simulation for lots of primitive polynomial, it seems to ...
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0answers
8 views

Which one will be better in Crout v/s Dolittle decomposition?

I recently read about the Cholesky , Crout and Dolittle decomposition. However, after studying Dolittle , I was wondering why is there a need for Crout decomposition to exist. I mean what upper hand ...
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1answer
49 views

Product of a Finite Number of Matrices with a Cosine Entry

Does any one know how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & a \\ b & 0 \end{pmatrix}\right)=2 $$ when $n$ is ...
1
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2answers
35 views

Why does similarity with a diagonal matrix imply that the Jordan normal form must also be diagonal?

If a matrix representation of a linear transformation is similar to a diagonal matrix, why does this imply that the Jordan normal form must also be diagonal?
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1answer
118 views

Proof using the Rank Theorem

Let $v$ and $w$ be non-zero column vectors in $\Bbb R^n$ and let $A = vw^t$ so that $A$ is an $n\times n$ matrix. Use the rank theorem to show dim Nul $A = n − 1$ Here Nul(A) represents the Null ...
2
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1answer
39 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 ...
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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 ...
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2answers
40 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 ...
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2answers
43 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. ...
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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} ...
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0answers
22 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} ...
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1answer
62 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 ...
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1answer
22 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} $ $ ...
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2answers
30 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 ...
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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 ): ...
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2answers
36 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?
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3answers
49 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 ...
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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
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1answer
41 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
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1answer
52 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 ...
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4answers
90 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 ...
0
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0answers
44 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 $$ (A-\lambda_1I)^\kappa $$ ...
9
votes
1answer
157 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 ...
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0answers
19 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 ...
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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 ...
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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
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1answer
40 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?
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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
16 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 ...
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2answers
34 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
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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
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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: ...
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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
39 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
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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) ...
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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
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1answer
47 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
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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 ...