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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271 views

Find the sub determinant of a matrix

If I have the matrix $A(x)$ which is $5 x 5$ and I need to find the $t x t$ sub-determinants of $A(x)$ for $t = 1$ to $t = 5$, how do I do this? $A(x)$ = \begin{pmatrix} (x-1) & 0 & 0 & ...
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2answers
89 views

Variant of Cholesky Decomposition: solve $B^TB=A$ for general square matrix $A$

Choelsky Decomposition allows us to decompose a Hermitian Positive Definite Matrix $A$ as $A=LL^*$, and $L$ is guaranteed to be lower-triangular. My Question: If $A$ is non-Hermitian, but still ...
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1answer
77 views

Determining whether binary matrix B is obtainable from binary matrix A via row and column permutations

Say you have two binary (i.e., (0, 1) ) m x n matrices A and B. Their row and column sums match up - i.e., for each attained row (column) sum k in A, there are the same number of rows (columns) with ...
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69 views

how to write QR algorithm into one equation to represent?

http://en.wikipedia.org/wiki/QR_algorithm is it possible to write it in one equation
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204 views

Norms of idempotent matrices

So if I have a random idempotent matrix (a matrix that when squared equals itself) how do I go about calculating its 2-norm? I know that a idempotent matrix has eigenvalues of only 0 or 1 and I know ...
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1answer
87 views

Optimize matrix multiplications

Given: vectors $v1, v2$ $(nx1)$ where entries in each vector are in the interval $[0,1]$. $v1$ and $v2$ can be sparse or dense a dense symmetric matrix $M$ $(nxn)$ (actually a logic matrix where ...
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114 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
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1answer
191 views

Example of Matrix in Reduced Row Echelon Form

I'm struggling with this question and can't seem to come up with an example: Give an example of a linear system (augmented matrix form) that has: reduced row echelon form consistent 3 equations 1 ...
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40 views

Problem related to a matrix

Taking $M$ to be of the form \begin{pmatrix} a &b &c \\ d & e & f\\ g& h & i \end{pmatrix} we get (from the $2$ given conditions) $6$ equations whereas total number of ...
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245 views

Solving a Conic Matrix given these Equations

Given a conic $\Gamma$ that has the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, $\Gamma$ can be represented by the symmetric matrix $$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ...
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228 views

A set of Quadratic equation, any good algorithm?

now I'm doing my research on filter design and I'm stuck in a mathematic problems. I want to solve the following equation: ...
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1answer
193 views

Rotating a line segment in 3D to a prescribed orientation

I have a general line segment with endpoints $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ referenced to a 3D Cartesian coordinate frame E. I wish to rotate this coordinate fram E to a new coordinate system F ...
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1answer
103 views

For a first order inhomogenous system of linear differential equations, what is a good way of defining resonance?

I apologize for the title being slightly unclear (at least to me it seems so), so if anyone has a better suggestion feel free to change it. Anyways, for example, when dealing with a second order ...
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1answer
168 views

Eigenvector of A to given Eigenvalue which requires row swapping to get reduced echelon form

Given the Matrix $$A = \left(\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -3 & -3 \end{matrix}\right)$$ calculate the eigenvalues and the corresponding eigenvectors of ...
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1answer
163 views

Verifying Orthogonality of Eigenvectors

How do you 'verify' the orthogonality of the eigenvectors of a matrix, let's say ${\bf A}$ , for example? I came across the result that a matrix ${\bf A}$ has orthogonal eigenvectors if ${\bf ...
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1answer
338 views

Derivative of ratio of functions, $f,g:\mathbb{R}^N\to\mathbb{R}$ with respect to a vector?

This is fairly simple, but my matrix calculus is not that strong. Given two functions, $f:\mathbb{R}^N\to\mathbb{R}$, $g:\mathbb{R}^N\to\mathbb{R}$, and $x\in\mathbb{R}^N$, how do I compute the ...
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1answer
99 views

norm of a matrix ( which norm have to use ?)

I need to find the norm of the matrix $$ A=\left( \begin{array}{cc} e^{-x} \cos( \sin x) & e^{-x} \sin ( \sin x) \\ -e^{-x} \sin ( \sin x) & e^{-x} \cos (\sin x) \end{array} \right) $$ Here ...
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1answer
432 views

Lower bound on norm of product of two matrices

Let $\vert \vert . \vert \vert$ be the 2-norm. Since this norm is submultiplicative, we know that for any two square matrices $A, B \in \mathbb{R}^{n \times n}$, $$ \vert \vert A B \vert \vert \leq ...
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1answer
233 views

LU decomposition with pivoting

I have to solve system of linear algebraic equations AX=B, where A is two-dimensional matrix and all elements of main diagonal are equal to ...
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1answer
52 views

Are the entries of this matrix expression positive?

Suppose $M$ is a square matrix with full rank. If $v$ and $w$ are column vectors, then the expression $$M^Tvw^TM =: A$$ is a matrix. Under what assumptions on $v$ and $w$ can we say that $A$ has ...
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1answer
110 views

Matrix solving problem

Hi there math experts. I have the following matrix: $$ \begin{equation} \begin{pmatrix} -1x & 0y & 0 & 0 & 0.004 & 0\\ -1x & -1y & 0 & 0 & 0.001 & 0 ...
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2answers
76 views

Math question using matrices

I have the following system: $$\left\{\begin{array}{cccccccc} 2x&+&3y&+&z&-&3v&=&2 \\ x&-&y&+&2z&+&v&=&0\\ ...
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1answer
38 views

Stabilization property of an operator in a finite-dimensional vector space?

Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$. (a) How do we prove the stabilization ...
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1answer
30 views

If $A\sigma=\sigma\tau$, does $[\tau]_B=A$ for some basis $B$?

I was trying to figure out the following at work today. Suppose $\tau\in\mathscr{L}(V)$, for $V$ an $n$-dimensional vector space over a field $F$. Let $A\in M_n(F)$, and $\sigma\colon V\to F^n$ be an ...
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145 views

Input-output economics and stability of general equilibrium

Here, I will start with a simple expression for an input–output system with $x(t)$ representing the vector of outputs and $A$ the input–output matrix. Then, the simplest possible linear ...
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1answer
2k views

Find the Matrix A of the Linear Transformation

Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix $A$ should be a $2 \times2$ matrix, should I be ...
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2answers
524 views

non-symmetric positive definite matrix!?

Is symmetry a necessary condition for positive (or negative) definiteness? If not: It can be proved that if $\mathbf{A}:(m\times m)$ is a square (non-symmetric) matrix, then $$ ...
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2answers
1k views

why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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1answer
186 views

2-rank of symmetric matrix

If $A$ is a symmetric integral matrix with zero diagonal, then I want to prove $2-rank(A)$ (i.e. the dimension of $C_A$ ) is even? $2-rank(A)$ means dimension A on field $\mathbb{F}_2$.
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1answer
18 views

Efficient inverse of $c'Cc$

Given that I have two matrix $c$ and $C$, which are $m \times n$ and $m \times m$ respectively with $m << n$, is there an efficient method for finding $(c'Cc)^{-1}$ that does not involve ...
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1answer
116 views

Derive Rigid Transform Matrix from Axes and Origin

I'm trying to derive the matrix of a rigid transform to map between two coordinate spaces. I have the origin and the axis directions of the target coordinate space in terms of the known coordinate ...
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1answer
99 views

Is f diagonalizable? Justify your answer

$f\begin{bmatrix}x\\\\y\\\\z\end{bmatrix}=\begin{bmatrix}-5x-y+3z\\\\-18x-3y+9z\\\\-16x-3y+9z\end{bmatrix}$ Here is what I have so far Let ...
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2answers
1k views

prove matrix norm equivalence

Given $A \in R^{m\times n}$, I need to prove: $$||A||_2 \le \sqrt {m}||A||_\infty$$ I have tried a number of things and I just cant seem to get it to work. Also, I need to prove: $$||A||_2 \le ...
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1answer
114 views

Generating random commuting hermitian matrices

How can I generate random commuting hermitian matrices ? EDIT: Another question: given a certain hermitian matrix, how can I generate a random hermitian matrix which commutes with it?
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1answer
85 views

Trace manipulation

I have matrix difference equation (Riccati equation): $$ X(k+1) = Q + F\left(I - X(k)H^{T}\left(HX(k)H^{T} + R\right)^{-1}H\right)X(k)F^{T}. $$ I have to work on the trace of this matrix. Is there ...
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1answer
386 views

Find the basis of eigenvalues

$$\begin{bmatrix} a\\\\b\end{bmatrix}\longmapsto \begin{bmatrix}-3&1\\\\0&2\end{bmatrix} \begin{bmatrix} a\\\\b\end{bmatrix}$$ Use the characteristic polynomail to find all eigenvalues for ...
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1answer
125 views

Linear Algebra Recreational Problem

For each positive integer $k$, find the smallest number $n_k$ for which there exist real $n_k$ by $ n_k$ matrices $A_1; A_2; ....; A_k$ such that all of the following conditions hold: $$ \text{ 1. } ...
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1answer
59 views

Values for entries of a matrix to have full row rank.

We are given the following matrix: $$ A=\begin{pmatrix} b_1 & \lambda_1b_1 & \lambda_1^2b_1 & \ldots &\lambda_1^{n-1}b_1 \\ b_2 & \lambda_2b_2 & ...
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1answer
99 views

Prove equivalence between the determinant of a matrix and the product of specific submatrices

Proposition Let $A\in\mathfrak{M}_{(m+n)\times(m+n)}(\mathbb{K})$, $B\in\mathfrak{M}_{n}(\mathbb{K})$, $D\in\mathfrak{M}_{m}(\mathbb{K})$. If $$A=\left(\begin{array}{cc}B& C\\0 & D ...
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1answer
474 views

Using permutation matrix to get LU-Factorization with $A=UL$

Let $Q$ be the $n$x$n$ permutation matrix $$Q= \begin{bmatrix} 0&0&...&0&1\\ 0&0&...&1&0\\ .& \\ .&\\ .&\\ 0&0&...&0&0\\ ...
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2answers
101 views

Calculate the rank of matrix $B-C$ while $AB=AC$ and $\operatorname{rank}(A) = r$?

$A$ is an $n\times n$ matrix and rank$(A)=r$.,$B,C$ are both $n \times n$ matrices and $AB=AC.$ Calculate the maximun possible rank of the matrix $(B-C)$. This question is a part of my homework in ...
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1answer
44 views

Does spectral normalization preserve eigenvalue ratios?

Let $A$ be a non-negative square matrix. Normalize $A$ by its spectral radius $\sigma(A)$, and call it $A_2 = A/\sigma(A)$. Does this normalization preserve the ratio between the two largest ...
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1answer
47 views

what can a matrix be similar to if and only if there exists a generalized eigenvector?

Let $M$ be a $n$ by $n$ matrix over a field $F$. $M$ is diagonizable, i.e. $M=P DP^{-1}$ for some invertible matrix $P$ and some diagonal matrix $D$, if and only if there exists an eigenbasis. I ...
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1answer
454 views

Finding generalized eigenbasis

For a complex square matrix $M$, a maximal set of linearly independent eigenvectors for an eigenvalue $\lambda$ is determined by solving $$ (M - \lambda I) x = 0. $$ for a basis in the solution ...
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1answer
189 views

operator norm and spectral radius

is it true that the operator norm of a matrix $A$ is smaller than 1 if its spectral radius $\rho(A)$ is smaller than 1? many thanks for any help, it is much appreciated!
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1answer
49 views

Given a matrix $A \in M_n(\mathbb{R})$, can we find two orthogonal matrix satisfy that$O_1AO_2$ is a diagonal matrix

Assume $A\in M_n(\mathbb{R})$ and $\det(A)\not=0$, is there existing two orthogonal matrix $O_1$,$O_2$ that satisfy $$O_1AO_2=\begin{pmatrix}\lambda_1 & & & \cr & \lambda_2 & ...
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1answer
96 views

Explanation of easy statement regarding derivative and Jacobian needed

Let $\Phi:S \to T$ be a map between surfaces in $\mathbb{R}^n$. What precisely does this mean: Let $\text{det}(\mathbf{D}_S \Phi(.))$ denote the Jacobian determinant of the matrix representation ...
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1answer
126 views

Finding the scalar derivative of a matrix product

I'm trying to find $$\frac{\partial}{\partial \lambda}y^T \left(\sigma^2 I + \lambda^{-1}K_{\theta}^{-1}\right)^{-1}y$$ where $y \in \mathbb{R^n}$ is fixed, $\lambda \in \mathbb{R}$ and ...
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1answer
158 views

Calculating the product of tridiagonal matrix times its transpose

Denote by tridiag($a$,$b$,$c$) the tridiagonal matrix of size $n \times n$ with diagonal elements $b = (b_1, \ldots,b_n)$. Let $a = (a_1, \ldots, a_{n-1})$ and $c = (c_1,\ldots,c_{n-1})$ be the ...
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2answers
701 views

Moore-Penrose pseudo inverse algorithm implementation in Matlab

I am searching for a Matlab implementation of the Moore-Penrose algorithm (convertable to C++) computing pseudo-inverse matrix. I tried several algorithms, "Fast Computation of Moore-Penrose Inverse ...