Articles of linear algebra

How to find expected solved variables from a linear equation

I have a difficult problem about probability that needs your help. Let $(x_1,x_2,x_3,x_4)$ be four unknown variables. From these variables, we can created a set of equations as follows: $$ \begin{pmatrix} x_1 & & & \\ & x_2 & & \\ & & x_3 & \\ & & & x_4\\ x_1 +x_2& & & \\ […]

Question about a projective module in a direct sum

I have a question about my method of proof for proving a simple fact about projective modules. I have a feeling my idea is wrong and I was hoping some one could point out where the mistake is. Let $R$ be a commutative ring with identity. Let $P$ be a projective $R$-module. Prove that there […]

Does this construction yield a basis for $\mathbb{R}^{m \times n}$ from a basis for $\mathbb{R}^m$ and $\mathbb{R}^n$?

Given a basis $(X_1, X_2, \ldots, X_m)$ for $\mathbb{R}^m$ and $(Y_1, Y_2, \ldots, Y_n)$ for $\mathbb{R}^n$, do the $mn$ matrices $X_iY_j^t$ form a bsis for the vector space $\mathbb{R}^{m \times n}$ of all $m \times n$ matrices? My initial guess is yes based on experimenting with the bases $\left\{(1, 0, 0), (0, 1, 0), (0, […]

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the symmetric semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes $||X||$ (Frobenius norm) and simultaneously fullfills $$ \Sigma_1+X=\Sigma_2+Y~~,$$ where $Y\in \textrm{SPD}^n$ and $\Sigma_1$ and $\Sigma_2$ are given, symmetric, and in $\text{PD}^n$. I suspect this problem is symmetric between $X$ and […]

Methods to solve a system of many Ax=B equations using least-squares

I am working with a force measurement instrument which needs calibration via a calibration matrix. For each of a set of controlled measurements I have a vector $k$ of three known, independent values, and a corresponding vector $m$ containing three measured values, that due to instrument constraints end up being non-orthogonal. From each measurement, I […]

Positive semidefinite but non diagonalizable real matrix – proof real parts of eigenvalues are non-negative

I have a question about positive semidefinite matrices that are non diagonalizable. Example: \begin{equation} A= \left(\begin{array}{cc} 2 & 1\\ 0 & 2\\ \end{array}\right) \end{equation} Clearly the (real part of the) eigenvalues of $A$ are non-negative. But how do I prove in general that the real part of the Eigenvalues of a positive semi-definite real matrix […]

$\dim(\ker\varphi\cap\ker\psi)=n-2$ proof

Let $V$ be a vector space of dimension $n$ over the field $K$. Let $\psi,\varphi$ be two non-zero functionals on $V$. Assume that there is no element $c\in K$,$c\neq 0$ such that $\psi=c\varphi$, Show that: $(\ker\varphi)\:\cap\:(\ker\psi)$ has dimension $n-2$ I know $\dim V=\dim Im+\dim \ker$, in which $Im$ is image. We can apply this equation […]

Prove that $A$ is similar to $A^n$ based on A's Jordan form

Let $A = \begin{bmatrix}1&-3&0&3\\-2&-6&0&13\\0&-3&1&3\\-1&-4&0&8\end{bmatrix}$, Prove that $A$ is similar to $A^n$ for each $n>0$. I found that the characteristic polynomial of $A$ is $(t-1)^4$, and the minimal polynomial is $(t-1)^3$. And the Jordan form of $A$ is \begin{bmatrix}1&1&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{bmatrix} I guess the key to solve this is to use the fact that two matrices are similar […]

Define $L(A) = A^T,$ for $A \in M_n(\mathbb{C}).$ Prove $L$ is diagonalizable and find eigenvalues

Let $L:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be defined by $L(A) = A^T,$ where $A^T$ is the transpose of $A$ and $M_n(\mathbb{C})$ is the space of all $n \times n$ matrices with complex entries. Prove that $L$ is diagonalizable and find the eignevalues of $L.$ I worked out a general $L$ that was $2 \times 2$ (might’ve found […]

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 […]