Articles of linear algebra

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if parabola and $<0$ if it is an ellipse. But my confusion is that they do not include pair of straight lines […]

3 nonzero distinct eigenvalues, part 2

This is an attempt to generalize the answer to a previous question Consider the $n \times n$ matrix $$A=\left[ \begin{array}{cccc} 0 & \frac{1}{n-1} & … & \frac{1}{n-1} \\ 1 & 0 & & 0 \\ \vdots & & \ddots & \\ 1 & 0 & & 0% \end{array}% \right] $$ (A has $n-1$ entries equal […]

projective geometry hyperplane

For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb K^n\rightarrow\mathbb KP^n, \tau_j(x_1,\ldots,x_n):=[x_1:\ldots:x_j:1:x_{j+1}:\ldots:x_n]$, where $e_j\in\mathbb K^{n+1}$ the $j’$th unit vector is. Now I come to my question; How can I show that the images of $\tau_j$ overlay whole $\mathbb KP^n$ or mathematically spoken: $\mathbb KP^n=\bigcup_{j=0}^n \tau_j(\mathbb K^n)$ I think […]

interpreting the effect of transpose in the normal equations

I have a question about the normal equation. $A$ an $m\times n$ matrix with trivial nullspace, $y$ a vector outside the range of $A$. The vector $x$ that minimizes $|| Ax – y ||^2$ is the solution to $A^*Az = A^*y$. How can I interpret what $A^*$ is doing? Why does multiplying $Ax = y$ […]

Finding eigenvalues and basis for linear transformation $T: P_{100} \to P_{100}$

Consider the linear transformation $T: P_{100} \to P_{100}$ given by $ T(p(t)) = p(t) + p(2-t) $ Find all eigenvalues and a basis for each eigenspace of T. So a standard basis for the $P_{100}$ is {$1,t,t^2,t^3,…,t^{100}$} $T(1) = 1 + 1 = 2$ $T(t) = t + (2-t) = 2$ $T(t^2) = t^2 + […]

Diagonalizing symmetric real bilinear form

I am given the following symmetric matrix: $$ A=\begin{pmatrix} 1 & 2 & 0 & 1\\ 2 & 0 & 3 & 0\\ 0 & 3 & -1 & 1\\ 1 & 0 & 1 & 4\\ \end{pmatrix}\in M_4(\Bbb R) $$ Let $f\in Bil(V), f(u,v)=u^tAv.$ I want to find a base $B \subset \Bbb R^4$ […]

Making a complex inner product symmetric

Let $(V, (\cdot, \cdot))$ be a complex inner product space, say a space of complex-valued functions, with $(\cdot, \cdot)$ linear in the second position and sesquilinear in the first. Assume that $V$ is closed under conjugation. I want to prove/disprove that the function $f\colon V\times V\to \mathbb{C}$ defined by $f(x, y)=(\bar{x}, y)$ is symmetric. If […]

Why is the geometric multiplicity of an eigen value equal to number of jordan blocks corresponding to it?

Geometric multiplicity of an eigen value is = $ dim$ $null$ $(A -λ I)$. $——-(1)$ Suppose A is in jordan normal form and has two Jordan forms with eigen value λ, one of size $2 X2$ and other of size $3X3$. Then, why is $ dim$ $null$ $(A -λ I)$ necessarily = $2 ??$ (i.e. […]

Solve the matrix equation $X = AX^T + B$ for $X$

Consider the matrix equation \begin{equation} X=AX^T+B, \end{equation} where $X$ is an unknown square matrix. Is it possible to solve it analytically? Moreover, can a general solution be written down in terms of the matrices $A$ and $B$?

Total number of subspaces of $\mathbb F_2^n$

I’m interested in getting a lower bound (or an exact number) on the number of subspaces of $\mathbb F_2^n$ over $\mathbb F_2$ – I’ll denote this number by $S(n)$. The number of subspaces of dimension $1\le k \le n$ is given by $$S(n,k) := \prod_{i=0}^{k-1} \frac{2^n – 2^i}{2^k – 2^i} $$ which isn’t too hard […]