Articles of matrices

How to solve this system of linear equations

$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 & e_4\\ b_4 & d_2 & a_3 & e_4\end{smallmatrix}\right)$$ All of the equations equal to 26; augmented, the matrix […]

Proving a given formula for projection matrix

In an $n$-dimensional inner product space $V$, I have $k$ ($k\le n$) linearly independent vectors $\{b_1,b_2,\cdots,b_k\}$ spanning a subspace $U$. The $k$ vectors need not be orthogonal. Then I was told that the projection of an arbitrary vector $c$ onto $U$ is given by $$P_Uc=A(A^TA)^{-1}A^Tc$$ where $A$ is the $n\times k$ matrix with column vectors […]

Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if the upper right zero matrix would be replaced with matrix C, would it still be true, that always $\ rank(X)= \ rank(A)+\ rank(B)$.

Solution of $A^\top M A=M$ for all $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can’t find out it there are other such matrices and if so, how to find them. Necessarily, $A$ is invertible since $\det(A)^2\det(M)=\det(M)\neq 0$ and more precisely, $\det A=\pm 1$. But […]

Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert Y\rVert_1$ denotes sum of absolute values of matrix entries ($\lambda \gt 0 $). $\lVert X\rVert_*$ denotes the nuclear norm of a matrix (sum of its singular […]

Properties of matrices changing with the parity of matrix dimension

Lately I encountered a situation when particular property of a real square matrix depended on the parity of matrix dimension, namely: for even dimension from equality of adjugates of invertible matrices we can infer about equality of matrices, for odd dimension there is no equivalence ( i.e. matrix function $B=\text{adj}(A)$ is bijective in the set […]

Prove that for a real matrix $A$, $\ker(A) = \ker(A^TA)$

So clearly the kernel of $A$ is contained within the kernel of $A^TA$, since $$A^T(A\vec{x}) = \vec{0} \Rightarrow A^T(\vec{0}) = \vec{0}$$. Now we need to show that the kernel of $A^TA$ is contained within the kernel of $A$. So suppose we have a $\vec{x} \in \ker(A^TA)$ so that $(A^TA)\vec{x} = \vec{0}$. How can we show […]

Is this determinant identity true?

I simulated the following $$\det(I+[A|B][A|B]^*)\geq\det(I+[B][B]^*)$$ and every time I get a true result. So how can I prove this statement? Here $[A|B]$ is matrix augmentation. $I$ is the identity matrix, $A,B$ are complex valued matrices and $^*$ is the conjugate transpose operation. In my first simulation I used randomly generated complex square matrices so I […]

How to transform (rotate) this hyperbola?

Given this hyperbola $x_1^2-x_2^2=1$, how do I transform it into $y_1y_2=1$? When I factor the first equation I get $(x_1+x_2)(x_1-x_2)=1$, so I thought $y_1=(x_1+x_2)$ and $y_2=(x_1-x_2)$. Therefore the matrix must be $ \begin{pmatrix} 1&1\\1&-1 \end{pmatrix} $. But in maple it is just drawn as a straight line. However, it should be a rotated hyperbola. Is […]

How find this matrix has eigenvalues $\lambda_{j}=4\sin^2{\dfrac{j\pi}{2(n+1)}}$

Show that the $n\times n$ tridiagonal matrix $$A=\begin{bmatrix} 2&-1&0&0&0\\ -1&2&-1&0&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&0&-1&2&-1\\ 0&0&0&-1&2 \end{bmatrix} $$ has the eigenvalues $$\lambda_{j}=4\sin^2{\dfrac{j\pi}{2(n+1)}},j=1,2,\cdots,n$$