Articles of linear algebra

set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

The set of almost complex structures on $\mathbb R^{2n}$ is given by $$ M_n = \frac{GL(2n,\mathbb R)}{GL(n,\mathbb C)} = \mathcal C_+ \sqcup \mathcal C_-,$$ taking into account that $\det = \pm 1$ gives two disjoint sets. How do we show that $M_2=S^2 \sqcup S^2$? Moreover, I’m a bit confused by dimensions: how does $\dim_{\mathbb R}M_2=8$ […]

Matrix for rotation around a vector

I’m trying to figure out the general form for the matrix (let’s say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards the origin and rotate counterclockwise). This is inspired by a similar problem which asked me to find the matrix for a […]

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler’s proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in “Linear Algebra Done Right”). In particular, it’s his maneuver in the last set of displayed equations where he substitutes the linear operator $T$ for the complex variable $z$. See below. What […]

REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?

This question already has an answer here: System of linear equations having a real solution has also a rational solution. 1 answer

If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator

Let $V$ be a finite-dimensional inner product space over $\mathbb{C}$ and $T: V \to V$ a linear transformation. Show that if every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal. We need to show that $\forall v \in V,\ TT^*v=T^*Tv$. I started by picking $v$ to be an eigenvector […]

If $T\alpha=c\alpha$, then there is a non-zero linear functional $f$ on $V$ such that $T^{t}f=cf$

I am self-studying Hoffman and Kunze’s book Linear Algebra. This is exercise $4$ from page $115$. It is in the section of The transpose of a Linear Transformation. Let $V$ be a finite-dimensional vector space over the field $\mathbb{F}$ and let $T$ be a linear operator on $V$. Let $c$ be a scalar and suppose […]

Maximizing a quadratic function subject to $\| x \|_2 \le 1$

Consider the $n$-dimensional quadratically constrained quadratic optimization problem $$\begin{array}{ll} \text{maximize} & \frac12 x^T A x + b^T x\\ \text{subject to} & \| x \|_2 \le 1\end{array}$$ where $A$ is a symmetric $n\times n$ matrix that may be indefinite. Given the symmetry of the constraint, is there a nice closed-form solution, perhaps in terms of the […]

Stuff which squares to $-1$ in the quaternions, thinking geometrically.

How can we think of the set$$\{x \in \mathbb{H} : x^2 = -1\}$$geometrically? Is this set finite or infinite? Are there some more geometric ways of thinking about than meets the eye? Here, $\mathbb{H}$ denotes the quaternions.

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ where $I_{n}$ is the identity matrix. This problem is from china (xixi) test question. It is said one can use the equation: $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)?$$ But I can’t use this to prove it. Can you help me? Thank you

What are some alternative definitions of vector addition and scalar multiplication?

While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren’t necessarily what we normally call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commutativity of addition, etc.). Is there any widely used vector space […]