Articles of linear algebra

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 […]

Rotation matrix in terms of dot products.

Let us suppose we have unit vectors $u, v \in \mathbb{R}^m$ with $u \neq v.$ Let $T(u,v)$ denote the unique rotation of $\mathbb{R}^m$ carrying $u$ to $v$ and which is the identity on the orthogonal complement of $\text{Span}(u,v).$I have found the claim that $T(u,v)$ can be defined by the formula: $$T(u,v)x = x – \dfrac{(u+v) […]

There exists a vector $c\in C$ with $c\cdot b=1$

Let $n$ be a positive integer, and $A$ be the set of all non-zero vectors of the form $(e_1,e_2,\dots,e_n)$, where $ e_i\in\{0,1\}$. So $|A|=2^n-1$. Let $B$ be a proper subset of $A$. Does there always exists a subset $C\subseteq A$ of size at most $n-1$ such that for any vector in $b\in B$, there exists […]

The number of subspace in a finite field

How to prove this conclusion If V is a vector space of dimension n and F is a finite field with q elements then number of subspace of dim k is

Difference between the algebraic and topological dual of a topological vector space?

What is the difference between the algebraic and the topological dual of a topological vector space, such as for example the Euclidean space $\mathbb{H}$? I am interested in intuitive as well as in detailled technical answers.

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 $$\lambda_{j}=4\sin^2{\dfrac{j\pi}{2(n+1)}},j=1,2,\cdots,n$$

Normal operator matrix norm

I have some troubles to show that the operator norm of a normal operator is always equal to its largest eigenvalue, how can I proof this? Does anybody of you have a hint? My problem is, that I do not know where to use that this operator is normal?

Showing that $x^{\top}Ax$ is maximized at $\max \lambda(A)$ for symmetric $A$

I’m hoping that someone can help me fix my proof for the following theorem: Given a $n \times n$ symmetric matrix $A$, $$ \max_{x : ||x||_2 = 1} x^{\top}Ax = \max \lambda(A), $$ where $\max \lambda(A)$ is the maximum eigenvalue of $A$. My proof attempt: Let $A$ be a symmetric matrix with eigenvalue decomposition $A […]

What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?

Consider a full rank $n\times n$ symmetric matrix $A$ (coming from a set of physical measurements). I do an eigendecomposition of this matrix as $$A = E V E^T$$ Most of the eigenvalues are positive, while a few are negative but with much smaller magnitude compared to the maximum eigenvalue. I want to convert this […]