Articles of matrices

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 […]

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 $Ax = y$ […]

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. […]

Reference for trace/norm inequality

I’m looking for a reference for a matrix-norm inequality that I used in this answer, which has a few equivalent forms. I will use notation that applies to complex vector spaces with a sesquilinear inner product, but of course the same applies over real matrices. The statement is as follows: Take $A,B \in \Bbb F^{n […]

Finding the eigenvectors (& describing the eigenspace) of a Householder transformation matrix

If one is asked to find the eigenvector(s) for a Householder transformation matrix, but one is not given the values of or dimensions of the unit vector $u$. So if $H = I_n – 2uu^T$ where $I_n$ is the n x n identity matrix and has length/norm $||u||^2 = 1$. It can easily be shown […]

Question about creating $2\times 2$ covariance matrix with call option?

I’m completely stuck on how to do this problem. How can you go about calculating the variance of $Y$ and the covariance between $X$ and $Y$? I’m not sure how to use the information given to solve this problem. Let X be normal with mean zero and variance $\sigma^2$. Let $Y$ be a call option […]

Jordan Exchange/Pivot Operation formula for non pivot rows and columns?

I’m really confused with the Jordan Exchange (or pivot operation) for non pivot rows and columns. I apologize if this seems to be easily googled, but I’ve been struggling and I’m not sure what I’m doing wrong given the formulas. For instance, given the matrix \begin{bmatrix}1&4&3&-2\\-1&2&1&0\\ -4&2&0&2\end{bmatrix} I understand that for the first row, we […]

$\sum\limits_{j,k} {{a_{jk}}} {\bar u_j}{u_k}\mathop = \limits^? \int_0^1 | \sum\limits_j {{u_j}} {x^j}{|^2}x{\mkern 1mu} dx$

Let $A\in M_n$ and $U\in \mathbb{C}^n$ and $a_{jk}=\int_0^1 x^{j+k-1}\,dx$ Is this true that $\sum_{j,k} a_{jk}\bar u_ju_k=\int_0^1\Bigl|\sum_j u_jx^j\Bigr|^2x\,dx$

Linear transformation of a subspace

How can I calculate the image of a linear transformation of a subspace? Example: Given a subspace $A$ defined by $x + y + 2z=0$, and a linear transformation defined by the matrix $$M= \left( \begin{matrix} 1 & 2 & -1\\ 0 & 2 & 3\\ 1 & -1 & 1\\ \end{matrix}\right) $$ What is […]

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix $A$? Thank you very much!