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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5
votes
0answers
70 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
3
votes
2answers
157 views

the rank of an interesting matrix

Let $A$ be a square matrix whose off-diagonal entries $a_{i,j} \in (0,1)$ when $i \neq j$. The diagonal entries of $A$ are all 1s. I am wondering whether $A$ has a full rank.
2
votes
1answer
107 views

Jacobian of $A^{-1}b$

I need to calculate the Jacobian $\frac{df}{dx}$ of $f=A^{-1}b$ where $A$ and $b$ are a function of $x$, the variable towards to differentiate. I thought $$\frac{df}{dx} = \frac{dA^{-1}}{dx} b + ...
-2
votes
1answer
82 views

Spatial matrices [duplicate]

Possible Duplicate: Does a “cubic” matrix exist? There is some theory about spatial matrices ?, for example matrices of order $3\times 3\times 3 ,A=(a_{ijk}),i,j,k=1,2,3$ where ...
2
votes
0answers
461 views

Submatrix Notation

I'm looking through some computer science papers and I see some notation that I'm just not familiar with. Consider an 5 x 6 matrix $$G = \begin{pmatrix} a_{0,0} & a_{0,1} & ...
1
vote
1answer
120 views

Set of homomorphisms from discrete upper triangular group into continuous u.t. group

Let $G$ be the group $$ \begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24}\\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \end{pmatrix} $$ ...
11
votes
1answer
834 views

4 by 4 Matrix Puzzle

I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer? Determine the number of $4\times ...
1
vote
5answers
211 views

Independence of Rotation Matrix Definitions

I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix $$ A = \left( \begin{matrix} a_{11}& a_{12}& a_{13}\\ a_{21}& ...
2
votes
2answers
233 views

Measure to compare matrix $A$ and permuted matrix $B$

I have a matrix $A$ and matrix $B$ of same dimension. We generally use $||A-B||_F^2$ (Forbenius norm) to compare these two matrix how close they are to each other. Here we assume the $col_i$ of matrix ...
1
vote
1answer
145 views

Natural numbers of inverted matrix

My question is bit more mathematical then algorithmic. Let's say I have 4x4 natural numbers matrix (including 0), is there any inverted matrix which also contains only natural numbers to it? Thanks ...
30
votes
1answer
2k views

Is the following matrix invertible?

$$ \begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly its ...
2
votes
2answers
267 views

How does the Siamese method to construct any size of n-odd magic squares work?

A Magic Square of order n is an arrangement of $n^2$ numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. To ...
0
votes
1answer
262 views

Tracing diagonal numbers on a 2D grid or matrix.

I need to test a cell on a 2D grid/matrix to see if it's diagonal numbers have a power of 2 in them. Example: ...
2
votes
2answers
164 views

Show matrix $A+5B$ has an inverse with integer entries given the following conditions

Let $A$ and $B$ be 2×2 matrices with integer entries such that each of $A$, $A + B$, $A + 2B$, $A + 3B$, $A + 4B$ has an inverse with integer entries. Show that the same is true for $A + 5B$.
2
votes
1answer
307 views

Matrix Multiplication in 3 Dimensions [duplicate]

Possible Duplicate: Is there a 3-dimensional “matrix” by “matrix” product? Is matrix multiplication of 3-dimensional matrices defined? I cannot wrap my mind around ...
2
votes
3answers
842 views

3D to 2D rotation matrix

I have been trawling through this forum but am struggling to understand the maths a bit. Currently I have a 2D plane within a 3D space and I have the coordinates for them. I want to work on this 2D ...
0
votes
1answer
69 views

Higher variance implies larger spread?

Suppose a configuration of points is given in $X\in\mathbb{R}^{n\times 2}$. Given that the configuration has zero column means, the variance of each axis, $x$ and $y$, can be expressed as ...
-2
votes
1answer
154 views

$\mathrm{det}(A-B)\neq0$ if and only if $\mathrm{det}(A+B)\neq0$?

Is it true that $\mathrm{det}(A-B)\neq0$ if and only if $\mathrm{det}(A+B)\neq0$? (For $A,B$ real $n\times n$ matrices, say.)
1
vote
5answers
822 views

Suppose A is an n-by-n matrix with its diagonal entries are n and other entries are one. Find determinant of A.

For $n \geq 2$, find the determinant of $A_{n}=\begin{bmatrix} n & 1 & 1 &\ldots &1 \\ 1 & n & 1 &\ldots &1 \\ 1 & 1 & n &\ldots &1 \\ \vdots & ...
1
vote
2answers
268 views

Conjugate of matrix

Over an arbitrary ring $R$ with unit, is the matrix\begin{pmatrix} a & 0 & b & 0\\ 0 & 0 & 0 &0\\c & 0 & d &0\\ 0 &0&0&0 \end{pmatrix} conjugate ...
3
votes
1answer
735 views

are there any bounds on the eigenvalues of products of positive-semidefinite matrices?

I have real positive semidefinite matrices (symmetric) $A$ and $B$, both are $n \times n$. I am looking for upper bounds and lower bounds on the $m$-th largest eigenvalue of $AB$, in terms of the ...
4
votes
0answers
166 views

What is the exponential generating function of the inverse matrix of an integer triangle?

Let ${Z}$ denote an integer triangle (like Pascal's), ${Z}_{n,k}$ for $0\leq k \leq n$ and let $f$ be an exponential generating function for polynomials $p_{n}(x)$ with $[x^k]p_{n}(x)= Z_{n,k}$. ...
2
votes
1answer
101 views

Diagonalizing a matrix for producing a orthogonal set of functions.

Given a basis $B = \{g_0, g_1,\ldots , g_{nk}\} $, I wish to construct a set of $nk$ functions $S$ such that $\langle S_i,S_j \rangle = \delta_{i,j}$ (so that $S$ is an orthonormal set) where $\langle ...
-2
votes
2answers
569 views

find corresponding number in a range by index

There's a array: array(1,2,3,...12, 1,2,3,...9,9,10,...12, 1,2,3,...12, 1,2,3,...12, 1,2,3,...6,6,7,8,...12, 1,2,3,3,4,...12, ...); All numbers are in the range 1-12, one number maybe occur twice ...
0
votes
1answer
316 views

Matrix convergence — determine the converged matrix

I have a square matrix $A_{n \times n}$ whose elements are either 0 or 1. The matrix $A$ changes in response to different events (the elements always being 0 or 1). After a series of changes, it ...
4
votes
1answer
109 views

Show $A$ is “real-equivalent” to its transpose

The problem statement: Let $A$ be a real $9\times 9$ matrix with transpose $B$. Prove that the matrices are real equivalent in the following sense: There exists a real invertible $9\times 9$ matrix ...
16
votes
4answers
1k views

The arithmetic-geometric mean for symmetric positive definite matrices

A while back, I wanted to see if the notion of the arithmetic-geometric mean could be extended to a pair of symmetric positive definite matrices. (I considered positive definite matrices only since ...
2
votes
1answer
84 views

When is the difference of the “1” matrix and a positive semidefinite matrix positive semidefinite?

Consider a positive semidefinite matrix $A\in M_n(\mathbb{C})$. Let $E$ denote the matrix in $M_n(\mathbb{C})$ all of whose entries are $1$. What are some natural sufficient conditions for $E-A$ ...
0
votes
1answer
332 views

Calculate Covariance matrix

How to find a covariance matrix from given mean values. e.g. Given mean values m1= (1/4) (3 1 1)T and m2 = (1/4) (1 3 3)T
3
votes
2answers
204 views

Presentation of discrete upper triangular group

Let $G$ be the nilpotent Lie group consisting of matrices $$ \begin{pmatrix} 1 & a_{12} & \cdots & a_{1,n}\\ 0 & 1 & \ddots & \vdots\\ \vdots & \ddots & \ddots & ...
2
votes
1answer
102 views

Why does this equation converge to 1?

The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow ...
5
votes
1answer
111 views

2x2 Matrices and Differences of Fractions

Consider the difference of two arbitrary fractions, $\frac{a}{b}$ and $\frac{c}{d}$. $$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$ The numerator is the determinant of the 2x2 matrix $$ \left( ...
1
vote
1answer
128 views

Totally Uni-modular Matrices

A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
5
votes
1answer
108 views

Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?

In a paper (corollary 1, p.14) the following identity is used: Let g be a unitary matrix. Then: $$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$ Now my question is why this ...
1
vote
1answer
649 views

How to prove that a matrix is positive definite?

Let $L$ be a Laplacian matrix of a strong connected and balanced directed graph. Define $$ L^{s}=\frac{1}{2}\left( L+L^{T}\right) .$$ Let $D$ be a diagonal matrix with $$ D=\begin{bmatrix} d_{1} & ...
2
votes
3answers
89 views

Show matrix polynomials are equal

Let $A$ be a matrix with no repeated eigenvalues: $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}.$ Let $p(x)$ and $r(x)$ be two polynomials satisfying $$p(\lambda_{i})=r(\lambda_{i}) \text{ for } ...
0
votes
1answer
80 views

Trace of a $227\times 227$ matrix over $\mathbb{Z}_{227}$

well, I know that the trace is the negative of coefficient of $x^{226}$ of the characteristic polynomial the matrix, but I dont know how the Char.Poly looks like in this case.please give me some ...
3
votes
1answer
323 views

Necessary and sufficient conditions for a matrix to be a valid correlation matrix.

It's not too hard to see that any correlation matrix must have certain properties, such as all entries in the range -1 to 1, symmetric, positive semi-definite (excluding pathological cases like ...
0
votes
2answers
577 views

Find value of K in matrix

Find the value of "k" in the equation: $k\left(\begin{array}{cc}3 & -1 \\ 5 & -4\end{array}\right) = \left(\begin{array}{cc}-3/4 & 1/4 \\ -5/4 & 1\end{array}\right)$ do I multiply $3 ...
0
votes
1answer
140 views

The row- and column-sums of a nonengative matrix with spectral radius less than $1$

Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$? Edit: What if the matrix has positive ...
2
votes
6answers
294 views

Short matrix algebra question

$A$ is a square matrix, and there is a matrix $D$ such that $AD=I$. I need help to prove either (a) or (b) below: (a) Show that for each $\vec b$, $A\vec x=\vec b$ has a unique solution. OR (b) ...
0
votes
1answer
918 views

Are a square matrix's columns and rows either both(separately) linearly independent or both(separately) linearly dependent?

Prove or disprove: Given a square matrix $A$,the columns of $A$ are linearly independent iff. the rows of $A$ are linearly independent.
0
votes
1answer
184 views

Trying to understand the Jacobian, part I.

So I am trying to understand the Jacobian, as it relates to the transformation of random variables. The nuts and bolts are buried in calculus however. Now, I have been reading this paper here, and I ...
0
votes
0answers
71 views

Transform a point to a new space. How is it working?

Let us assume that you simply have a point: $(x_1, x_2).$ You also have a transformation $H,$ that maps this single point to a new point: $(y_1, y_2).$ So $y_1 = h_1(x_1, x_2),$ and $y_2 = h_2(x_1, ...
1
vote
1answer
71 views

homeomorphism of a subset of $GL_3(\mathbb{R})$ with $GL_2(\mathbb{R})$ and connectedness

Suppose I denote $G_3$ be the set of all $3\times 3$ matrices with positive determinant, and consider the map $\pi:G_3\rightarrow \mathbb{R}^3\setminus\{0\}$ define by $\pi(g)=ge_1$ where ...
0
votes
1answer
228 views

Completing a unitary matrix given a column

I am given a unit vector $e=1/\sqrt{n}(1,1,\ldots,1)'$ and the problem is to construct an $n\times n$ (real) unitary matrix $U$ which will contain $e$ as the last column. I understand that there ...
2
votes
2answers
860 views

Determine whether the set $H$ of all matrices form a subspace of $M_{2 \times 2}$

Determine if the set $Z$ of all matricies form $ \left[ \begin{array}{cc} a & b \\ 0 & d \end{array} \right] $ is a subspace of $M_{2 \times 2}$ (the set of all $2 \times 2$ matrices). % ...
1
vote
1answer
67 views

matrix inversion problem

Given a matrix $A$ and an identity matrix $E$, we can get the resultant matrix $X={(E-A)}^{-1}$. Now for a given diagonal matrix $D$, we would like to compute the matrix $Y={(E-DA)}^{-1}$. Is there ...
5
votes
3answers
580 views

Projection matrices

I have found these two apparently contradicting remarks about projection matrices: 1) A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection ...
1
vote
2answers
127 views

Any $X\in O^+(n)$ (orthogonal matrices with positive determinant) is the product of an even number of reflections?

Any $X\in O^+(n)$ (orthogonal matrices with positive determinant) is the product of an even number of reflection? I am not able to prove this. Please help.