Questions on the various algorithms used in linear algebra computations (matrix computations).

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1answer
69 views

Hessenberg Matrices

$$A=\begin{bmatrix} 2&3&4\\ 3&-5&5\\ 4&5&0\end{bmatrix}$$ Find a unitary matrix $Q$ such that $A = QHQ^{H}$, where $H$ is Hessenberg. I am having a little trouble ...
4
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1answer
220 views

The spectral norm of projection matrices

Given any $n \times n$ non-symmetric projection matrix $P$, i.e., $P^2 = P$ but $P^T \ne P$, is the spectral norm of $P$ bounded by a constant which is independent of the dimension $n$?
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1answer
92 views

Basic question about matrix algebra- notation

The representation $X=(I_p,0_{n-p\times p})$is confusing me. I get that $I_p$ is an identity matrix with $p$ rows and columns and the rest of the representation is confusing me. Can someone clarify ...
3
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1answer
130 views

Matrix trace based formulation of least-squares

How can the following function be represented in a matrix form using matrix trace? $||y-X\beta||^2 + \lambda \beta^T S \beta$ Note that $y, \beta$ are real vectors and $\lambda$ is a real scalar ...
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2answers
91 views

Why are the inverse results not equal?

I mat a problem when solving inverse of a matrix. Firstly, I use python numpy library to make it, by coding below: ...
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0answers
290 views

Weighted linear least squares parameter covariance

I am currently trying to figure out the parameter covariance for a weighted linear least squares problem where $$y = X\beta$$ The parameters for which my objective function is lowest are given by ...
3
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1answer
39 views

Arrange the computation of $f(x)$ so that the loss of significant figures can be reduced using $4$-digit decimal arithmetic

For the function $f(x)=\sqrt{x^2+1}-x$, have computed $f(100)=0$ using 4-digit decimal arithmetic (rounding after every intermediate calculation). The value $f(100) = 0.0049998750$ (to 8sf) is given ...
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0answers
22 views

How to construct an optimal subspace with 3 indices.

I have a 3-dimensional array that is potentially very large and I need to do quite a lot of operations with it. Is there a systematic way to choose a subspace of a certain size, such that the norm ...
2
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1answer
244 views

Numerical Analysis using MATLAB. Find the condition number $\mu$

Find the condition number $\mu = |A||A^{-1}|$ for the Hilbert Matrix $A$ using the uniform form. $A = \left( \begin{array}{cccc} 1 &\frac{1}{2} & \frac{1}{3} &\frac{1}{4} \\ \frac{1}{2} ...
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3answers
632 views

What are the real life applications of quadratic forms?

What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
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1answer
115 views

Transcendental function implementation using linear algebra techniques

I'm sure this has been answered, but I am not having much luck finding it. Is there a way to implement transcendental functions ($\sin(x)$, $\cos(x)$, $e^{x}$, $\ln(x)$, etc...) using techniques from ...
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1answer
190 views

affine set definition equivalence

How to show the following definitions are identical for an affine space: $C = p + W$ where $W$ is a subspace $p$ is a vector in $\mathbb{R}^n$, and $\lambda a + (1-\lambda) b$ is in $C$ for any $a$ ...
2
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1answer
89 views

Does the equation QX=B have solutions?

Given Q $\in \mathbb{R}^{n\times n} $ Orthogonal matrix and X,B $\in \mathbb{R}^{n\times n}$ generic matrixes. Demonstrate that the equation QX = B has solutions and provide an algorithm which ...
6
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1answer
138 views

small rank solution of a matrix equation

Consider the matrix equation $$AX-XA = R$$ where $A$ and $R$ are given square matrices such that $\operatorname{rank}(R)=r$. How to establish conditions (necessary, sufficient, or both) on $A$ and ...
2
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0answers
161 views

Orthonormal Matrix weighted regression

$Q$ is a rectangular matrix with orthonormal columns. A linear system composed of $$Qx= b$$ is really easy to solve as: $$Q'Q=I$$ hence: $$x=Q'b$$ Given that $Q$ is orthonormal can this be used to ...
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1answer
391 views

QR factorization for ridge regression

I am solving an overdetermined system of equations: $$Ax= b$$ Using QR factorization, we can solve this system easily by posing it as: $$Rx= Q'b$$ I would like to regularize my estimate of $x$. I ...
5
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0answers
203 views

numerical linear algebra tricks for repeated sums and inversions with symmetric positive-definite matrices

I'm doing the following procedure to get the max-likelihood estimate of a matrix-variate normal distribution from $r$ samples of matrices in $\mathbb{R}^{n \times p}$ (algorithm from Dutilleul ...
3
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1answer
641 views

Matrix Pseudo-Inverse using LU Decomposition?

What is the step by step numerical approach to calculate the pseudo-inverse of a matrix with M rows and N columns, using LU decomposition? So far, I have found this, but it uses singular value ...
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1answer
141 views

How to determine maximum angles between vectors?

I'm attempting to distribute vectors with the same origin with a maximum angle of separation. Then if given a set of vectors, I want to determine how far from maximum separation they are. For ...
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2answers
112 views

Lanczos vectors

I am trying to implement the Lanczos algorithm. If I implement it in Fortran or C, (i.e. in finite precision), will the vectors generated at each iteration still preserve their linear independence? ...
4
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1answer
193 views

Eigenvector of a sparse structured matrix corresponding to the eigenvalue 1

I have a matrix with the following sparsity pattern: $M = \begin{bmatrix} \ast &\ast &0 &0 &0 &0 &0 &0\\ 0 & 0 &\ast &\ast &0 &0 &0 &0 \\ 0 ...
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1answer
239 views

butcher tableau for given algorithm

Given $y'(t) = f(t,y(t))$ and the following algorithm: $$y_{n+\frac{1}{2}} = y_n + \frac{h}{2}f(t_n,y_n)$$ $$y_{n+1} = y_n + hf(t_n+\frac{h}{2},y_{n+\frac{1}{2}})$$ We should show that this can be ...
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2answers
73 views

convergence of newton algorithm when looking for roots of $f(x) = xe^x$

I want to show the convergence of newton's algorithm when calculating the root of $f(x)=xe^x$ using an $x_0 \geq 0$. The resulting recursion is $$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)} = x_k - ...
2
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1answer
323 views

householder transformation matrix

Hi could you help me with the following: Let A be the matrix $$\pmatrix{-2 & 1& 1 \\ -2& 2& 1\\2 &-2& 3 \\ }$$ with an eigenvalue $\lambda = 2$ and corresponding ...
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1answer
142 views

Conjugate gradient method

http://www.webpages.uidaho.edu/~barannyk/Teaching/hw7_Math432.pdf Can someone help with #2? This is NOT homework. This was a website for Fall 2011, but it is a very interesting question and I would ...
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1answer
60 views

multiplying rows to get great pivots for numerical $LR$ decomposition of invertible Matrix $A$

When calculating the $LR$ decomposition of an invertible Matrix $A$ with the numerical Gauss algorithm there can happen to be inaccuracies caused by the computers precision. An example is ...
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4answers
295 views

$A^T A$ eigenvalue bounding

Let $A$ be a square matrix with real entries. Is there anything like any eigenvalue of $A^tA \leq \max({1,\lambda^2})$ where $\lambda$ is an eigenvalue of $A$ and max is taken over all eigenvalues?
4
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1answer
96 views

solve $ y = (A+B^{-1})x $ for $x$

I wish to solve numerically for $x$, $$ y = (A+B^{-1})x $$ with $A, B$ positive definite. So, $$ x = (A+B^{-1})^{-1}y $$ I would like to avoid calculating $B^{-1}$ since that's generally bad. ...
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0answers
449 views

covariance matrix eigenvalues eigenvectors

Is there a probabilistic or analytical meaning of the eigenvalues/eigenvectors of covariance matrix of multivariate normal distribution? Thank you
2
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0answers
170 views

$l^1$ norm estimate for inverse of Vandermonde matrix

As title, I would like to know the known upper bound for the $l^1$ norm for inverse of Vandermonde matrix. A quick search gives this paper by Gautschi 40 years ago, but it deals with the infinity norm ...
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2answers
288 views

l_2 norm matrix eigenvalue equality

Hello how to show the following: $||A||_2$ = $\sqrt{ \text{largest eigenvalue of } A^{T}A}$ for any $m\times n$ matrix $A$. Thank you
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2answers
148 views

spectral radius convergence as infinity

Hello how to show the following fact: When the spectral radius of a matrix is less than $1$ then $B^n\to0$ as $n$ goes to infinity? Thank you!
1
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1answer
199 views

LU decomposition permutation matrix

Hi can you help me with the following; Let $A$ be an $n\times n$ matrix and have $LU$ decomposition with lower and upper triangular matrices. Let $P =\{e_n,e_{n-1},\ldots,e_1\}$ where $e_i$ is a ...
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0answers
114 views

Can antiunitary symmetry be used to calculate determinant of a matrix

Suppose I have some $N \times N$ complex matrix $A$, that commutes with some antiunitary operator $U$ that satisfies $U^2 =-1$. It can be shown that $\det(A)\ge 0$ , because for every eigenvector ...
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2answers
2k views

matrix similarity upper triangular matrix

How to show: Any matrix A with real or complex entries is similar to an upper triangular matrix M whose diagonal entries are the eigenvalue of A. Thank you!
2
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2answers
346 views

Condition number inequality

Let A be an invertible n x n matrix, How to show that: $K(A) \ge \dfrac{\|A\|}{\| B - A \|}$ where $K(A)$ is the condition number of the matrix $A$ and for any $B$ being an $n\times n$ singular ...
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0answers
44 views

Numerical linear algebra spectral norm limit.

Let $A \in \mathbb{R}^{m \times n}$ be of full rank. Consider $X_{k+1}=(2k-X_{k}A)X_{k}$, $X_0 = \alpha A^{T}$. Let $E_k = I-X_kA$, Deduce that if $||E_{0} ||_{2}<1$, then $lim_{k \rightarrow ...
2
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2answers
329 views

1st-order linear ODE with tridiagonal matrix. Efficient solutions?

I have a 1st-rder linear ODE system where the system is characterized by $A$. Given an initial state $x_0$, I want the state at some later time $t$, efficiently. $A$ happens to be a symmetric ...
3
votes
1answer
617 views

Trace Minimization of Covariance Matrix

Given a matrix X whose rows contain observations collected at some locations. Can someone explain how trace minimization of covariance matrix $XX^T$ can lead to orthogonal / mutually independent ...
1
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2answers
103 views

smallest eigenspace dimension symmetric

Hi would you help me with the following: Let $A = (a_{ij}) R^{n \times n}$ be a symmetric matrix satisfying: $a_{1i} \neq 0$; Sum of each row equals $0$ and each diagonal element is the sum of ...
5
votes
2answers
4k views

Power iteration smallest eigenvalue?

I need to write a program which computes the largest and the smallest (in terms of absolute value) eigenvalues using both Power Iteration and Inverse Iteration. I can find them using the Inverse ...
2
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0answers
55 views

Solution to pertubed linear system

Suppose one has the following system of linear equations $$(A + \Delta A) x = b$$ where $A$ and $\Delta A$ are large sparse matrices and $\Delta A$ is "small" compared to $A$, furthermore vector $x$ ...
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1answer
91 views

iteration convergence bounds with norm less than 1

Let $x_{k+1} = Bx_k + c$ where $B$ is $n \times n$ matrix $c$ is a vector. Assume $\|B\| \le \beta <1$ $\|x_k - x_{k-1}\| \le \varepsilon$ for some $k$ Show that $\| x - x_k\| \le ...
5
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1answer
157 views

real eigenvalue

Let matrix $A$ be $$\begin{bmatrix} -5& 1& 0& 0\\ a &2& 1 &0\\ 0& 1 &1 &1\\ 0 &0&1& 0 \end{bmatrix}$$ where $a$ is a constant between 1 ...
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1answer
657 views

norm less than 1 matrix theory

Let $A$ be any $n \times n$ matrix and $\| \cdot \|$ be the matrix norm induced by vector norm on $\mathbb{R}^n$ (Euclidean n-dimensional space). If $\|I - A\| < 1$, then show that $A$ is ...
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2answers
127 views

Matrix norm less than $1$ iteration

Is the following true always for a matrix norm $$\lVert AB\rVert \leqslant \lVert A\rVert \cdot \lVert B\rVert \text{ ?}$$ Related to this given $r$ is positive constant, $H$ is symmetric positive ...
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0answers
135 views

fixed point spectral radius

We have the following stationary matrix iteration $$x_{k+1} = Mx_k + c$$ where $M$ is nxn matrix and $c$ is a vector. Let $r(M)$ denote the spectral radius of $M$. Show that spectral radius ...
2
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2answers
227 views

least square problem normal equations

Can you give an example which shows that loss of information can occur in forming the normal equations. How is accuracy improved using iterative improvement? Thank you
2
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1answer
135 views

Cholesky factorization counterexample

Could you give a counter example of a symetric matrix for which the Cholesky factorization does not exist? Also why does any eigenvalue solver have to be iterative?
5
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3answers
162 views

how can a matrix vector product reduce to a scalar?

I have an Excel spreadsheet with the following formula (paraphrased): ...