Articles of linear algebra

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some $\textbf{x}=(x_0,\ldots,x_n)$ and this yields $\textbf{y}=(y_0,\ldots,y_n)$, where $y_i=f(x_i)$. We shuffle the elements in $\textbf{y}$. Then we give $\textbf{x}$ and the shuffled $\textbf{y}$ to an untrusted server. The […]

Eigenvectors of a skew-symmetric matrix

I’ve read that any real skew-symmetric matrix $A$, where $A^T = -A$, can be brought into block diagonal form $ A = Q \, \Sigma \, Q^T = \left( \begin{array}{ccccl} \vec{q}_1 & \vec{q}_2 & \vec{q}_3 & \vec{q}_4 & \dots \end{array} \right) \, \left( \begin{array}{ccccl} 0 & \lambda_1 & 0 & 0 & \dots \\ -\lambda_1 […]

form of symmetric matrix of rank one

The question is: Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one. I think we can easily prove that if $C$ has the form $C=aww^T$, then $C$ is symmetric and of rank one. Why […]

Blockwise Symmetric Matrix Determinant

This question arises from another one of mine, but separate enough that I feel it deserves its own thread. Wikipedia says that $$det\begin{bmatrix}A&B\\B &A \end{bmatrix} = det(A+B)det(A-B)$$ Regardless of whether or not A and B commute. Using the general formulation $$det\begin{bmatrix}A&B\\C &D \end{bmatrix} = det(A)det(D – CA^{-1}B)$$ We see that this becomes $$det(AD- ACA^{-1}B)$$ Or […]

Find the real vector $x$ which satisfies all this?

I got this after applying KKT conditions to an optimization problem. Let $\mathbf{h}$ be a given $N\times 1$ real vector. Let $\alpha$ be a real constant. We need to find $\lambda$ and the $N\times 1$ real vector $\mathbf{x}$ such that \begin{align} (\mathbf{h}\mathbf{h}^T-\lambda\mathbf{I})\mathbf{x}&=\alpha\mathbf{h} \\\ \lambda &\geq 0\\\ \lambda(\mathbf{x}^T\mathbf{x}-1) &= 0 \\\ \mathbf{x}^T\mathbf{x}&\leq1 \end{align}

$A \in Gl(n,K)$ if and only if $A$ is a product of elementary matrices.

I want to show the statement: $A \in GL(n,K)$ if and only if $A$ is a product of elementary matrices. I could show the reverse implication: Suppose $A=T_1…T_m$ where each $T_k$ is an elementary matrix. An elementary matrix can be of the form: 1) $T_{ij}$ the matrix that swaps the rows $i,j$ 2) $T_{\lambda i}$ […]

Determinant of skew- symmetric matrix

I have to prove that determinant of skew- symmetric matrix of odd order is zero and also that its adjoint doesnt exist. I am sorry if the question is duplicate or already exists.I am not getting any start.I study in Class 11 so please give the proof accordingly. Thanks!

Total Derivative and Multilinear Functions

So I’m starting to work through Spivak’s Calculus on Manifolds and I’m having a little trouble verifying some of the claims made in the book problems. To review: Given a function $\mathbf{f}:\mathbb{R}^{n}\to\mathbb{R}^m$, we say that $\mathbf{f}$ is differentiable at a point $\mathbf{a}=(a^{1},\ldots,a^{n})\in\mathbb{R}^n$ (considered as a $1\times n$ matrix) if there exists a linear transformation, $D\mathbf{f}(\mathbf{a}):\mathbb{R}^n\to\mathbb{R}^m$, […]

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear functional $\phi \colon M_n(\mathbb{C}) \rightarrow \mathbb{C}$ such that $\phi(Id)=1$ and $\phi(A^*A) \geq 0$, for each $A \in M_n(\mathbb{C})$. Show that $\phi$ must […]

Asymptotes and focus of a conic?

This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$ the matrices associated with the conic are: $$ A’=\left(\begin{array}{cccc} \frac{1}{2} & 1 & \frac{1}{2} \\ 1 & 1 & 3 \\ \frac{1}{2} & 3 & 1 \end{array}\right), $$ $$ A=\left(\begin{array}{cccc} 1 & 3 \\ 3 & 1 \end{array}\right), $$ His characteristic polynomial is: $p_A(\lambda) = \lambda^2-2\lambda-8$ A has eigenvalues discordant […]