Articles of matrices

Finding an orthogonal matrix with given absolute value

$\DeclareMathOperator{\Abs}{Abs}$Define the absolute value of a matrix $A = (a_{ij})$ by $$ \Abs(A) = \pmatrix{|a_{11}| & \cdots & |a_{1n}|\\ \vdots & \ddots & \vdots\\ |a_{n1}| & \cdots & |a_{nn}|} $$ Suppose that you are given the absolute value of some unknown orthogonal matrix and you are supposed to find and orthogonal matrix with this absolute […]

Transformation matrix: rotation in $\mathbb R^3$

Operator $\phi: \mathbb R^3 \to \mathbb R^3$ is rotation around line $p: x_1 – x_2 = 0,$ $ x_3=0$, $\phi (0,0,2) = (\sqrt2,-\sqrt2,0)$. I need to find transforamtion matrix $A$: $\phi(x)=Ax$ in standart base of $\mathbb R^3$. How do I do that?

Projection onto subspaces – point to line projection

In the following document about projection onto subspaces, the author is computing the transformation matrix to project a vector $b$ onto a line formed by vector $a$. Since the projected vector $p$ is on the $a$ line, therefore it can be expressed as $p = \bar{x}a$. The projection “line” which is the vector $b – […]

Accessing elements of packed symmetric distance matrix

Suppose you have a symmetric distance matrix A. For example A is 4*4 (the numbers above and on the left of matrix are the indices of elements between which the distance is measured, and we use only the lower triangle): 0 1 2 3 _____________ 0 |0 0 0 0 1 |a10 0 0 0 […]

Approximate largest (in quantity, not magnitude) eigenvalue and eigenvector of a matrix

I need to obtain an approximate expression for the eigenvector corresponding to the largest real eigenvalue of a matrix, as well as the largest eigenvalue. Note that I mean largest not in absolute value, but largest in real value. The matrix can be approximately diagonal in some cases, but not always. I have tried perturbation […]

Show adding rows to a non-singular square matrix will keep or increase its minimum singular value

I realize the following problem can be summarized as to show Adding rows to an $n \times n$ non-singular matrix will keep or increase its minimum singular values. Let $\bf A$ be an $m \times n$ matrix in $\Bbb C^{m\times n}$ with $m>n$ and the first $n$ rows of it being linear independent. Thus ${\mathbf{A}} […]

What does the superscript $t$ in this matrix addition problem mean?

I know matrix addition is pretty easy – for a problem like $$\begin{bmatrix} a&b\\c&d\end{bmatrix}+\begin{bmatrix}e&f\\g&h\end{bmatrix}$$ The solution is $$\begin{bmatrix} a+e&b+f\\c+g&d+h\end{bmatrix}$$ However, I am working through some problems in this pdf and I can’t seem to figure out what is meant by the following problem (pg 51, problem C10 part 3): Let $B=\begin{bmatrix} 3&2&1\\-2&-6&5\end{bmatrix}$ and $C=\begin{bmatrix}2&4\\4&0\\-2&2\end{bmatrix}$ Find […]

Number of $k\times n$ matrices of rank $k$

How can I determine the number of $k\times n$ matrices with entries in $\mathbb F_p$ with rank $k$ (of course $k<n$) The formula if $k=n$ is $(p^n-1)(p^n-p)\dots(p^n-p^{n-1})$, now how can I modify this ? If I pick any $k$ columns of $n$, and apply the formula, and let the $n-k$ columns be arbitrary, then this […]

How to prove this inequation of matrix norm?

Suppose a square matrix $A=(a_{ij})_{n\times n}$ is irreducible. It is given that there exits $i_0$ for $$\sum_{j=1}^{n}{|a_{i_0j}|}<\|A\|_{\infty}$$ Out goal is to prove: $$\rho(A)< \|A\|_{\infty}$$ I was stuck at the very begining. $\rho(A)$ is the spectral radius of A. I guess the essence is to prove the equality is not satisfied with the given condition. However […]

Relation of eigenvalues of a matrix $A^2$ , where $A$ is a matrix consisting cube roots of unity

Matrix $A$ is $$ \begin{pmatrix} 1 & 1 & 1 \\ 1 & w^2 & w \\ 1 & w & w^2 \\ \end{pmatrix} $$ Where $1,w,w^2$ are cube roots of unity. My problem is to obtain the result $|\lambda_1| + |\lambda_2| +|\lambda_3| \le 9$, where, $\lambda_1 , \lambda_2, \lambda_3$ are eigenvalues of matrix $A^2$. […]