# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### Linear recurrent sequences and matrices.

Let $k$ be a field, let $d$ be an integer greater than $1$, let $(v,x)\in k^d\times k^d$ and let $A\in k^{d\times d}$ be invertible. For all $n\in\mathbb{N}$, let define the following element of $k$:...
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### To show a set of projectors summing to the identity implies mutually orthogonal projectors

The general setting is the study of positive operator measures in quantum mechanics http://en.wikipedia.org/wiki/Positive_operator-valued_measure instead of the projector operator measures. Going ...
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### The trace functional and its scalar multiples [duplicate]

I am trying to solve the following problem: Show that the trace functional on $n \times n$ matrices is unique in the following sense. If $W$ is the space of $n \times n$ matrices over the field $F$ ...
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### A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=?$

A four square matrix and $A^5$(a raised to the power of 5)$=0$. Then $A^4=$ $I$(identity matrix) $-I$ $0$ $A$ My attempt: You can use the characteristic equation $$A^2-Tr(A)A+I_2\det{A}=O_2,$$ ...
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### Find the eigenvalues the block matrix $M=\begin{bmatrix}A+2D & A \\ A & D \end{bmatrix}$

Let $A$ be any square matrix with eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$ and $D$ is a diagonal matrix with entries $d_1,d_2,\cdots,d_n$, then how can one find the eigenvalues of the ...
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### Relationship between geometric multiplicity, algebraic multiplicity and left and right eigenvectors of a matrix

The following statement is from the book Matrix Analysis by Horn and Johnson. An eigenvalue λ with geometric multiplicity 1 can have algebraic multiplicity 2 or more, but this can happen only if ...
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### Regarding element-wise derivative of matrices

Let $X=(x_{ij})$ be a real n-by-n matrix where $x_{ij}$ are in a range such that $X$ is insvertible. Let $Y=X^{-1}=(y_{pq})$, and regard $y_{pq}=y_{pq} (x_{11},x_{12},\ldots,x_{nn})$ as a function of ...
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### Dumb question: Orthogonal complement of kernel = Row space

I'm really confused when trying to prove the following: $\mathrm{kern}(A)^\perp = \{y \in \mathbb{R}^n \mid y = z^T A, \ z \in \mathbb{R}^n \}$ The $\supseteq$ direction is easy: Let $z^T A \in$ ...
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### Is there a name for matrices that are symmetric along the cross diagonal? [duplicate]

Something like $$A= \begin{bmatrix} a & b & c\\ b & d & e\\ c & e & f \end{bmatrix}$$ would be a symmetric matrix because the values are reflected along the ...
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### Is this a matrix notation of standard error?

What is the matrix notation of the standard error? A friend is referencing the standard error as: $$SE^2=(XX^T)^{-1}\sigma^2$$ $$\sigma^2 = \frac{1}{n-p}\sum\limits_{i=1}^n |\hat{y_i}-y_i|$$ ...
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### Can one factor matrices?

I know that one can factor integers as a product of prime numbers. Is there an analog of it to matrices? Can we define prime matrices such that every matrix is a product of prime matrices? Is there ...
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### $AB-BA=I$ having no solutions

The question is from Artin's Algebra. If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions. I have no idea on how to tackle this question. I tried block ...
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### the rational canonical

let T$\in$ $\mathcal{L}$($\mathbb{Q^3}$,$\mathbb{Q^3}$) be given by $$T(v)= \left[ \matrix { 1&-1&-4 \\ 1&-1&-3 \\ -1&2&-2 } \right]v$$ . Find the rational canonical form ...
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### Minimizing $\|Ax\|_2$ subject to $\|x\|_2 = 1$

I have a Matlab program to estimate a vector $x$ from noisy measurements. I use the singular value decomposition (SVD) to solve the linear equation $Ax=0$ (where the number of equations is greater ...
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### 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: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
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### Points on which function is invertible?

$f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $f(x,y)\mapsto((x-y)^{2}+1, x-y^{3}-2)$ For which points is this function invertible? I calculated the Jacobian matrix, but what should I do next to get ...
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### Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
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### For matrices $C, D$, show that $(CD)^{100} \neq C^{100} D^{100}$

The question is to prove this is false: $(CD)^{100} = C^{100}\cdot D^{100}$, where $C$ and $D$ are matrices. I looked through my textbook and could not find a proof for this.
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### Matrix with all entries N

Is there a specific name for a matrix where all entries are the name number? I am writing a program where I want to be able to describe a matrix like this in the same way I would the identity matrix, ...
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### Matrix power relation to prove coefficients exist

Let M be a 2x2 Matrix and $n$ an integer $2\geq\ n$. Prove that there exist integers $a_n$ and $b_n$ such that $$M^n=a_nM+b_nI$$ To begin, what I did was to use the Cayley-Hamilton Theorem then use a ...
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### For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold?

For two square positive-semidefinite matrices $A$ and $B$, does the relation $|A|^{-\frac{1}{2}}|B|^{-\frac{1}{2}} = |AB|^{-\frac{1}{2}}$ hold? Here the absolute value signs are the determinant ...
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### Inverse of the sum of identiy matrix and a symmetric matrix

Is there a simple way to solve $(I + A) X = B$, where $I$ is the identity matrix, and $A$ is a symmetric matrix?
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### Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
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### Is a correlation matrix with positive determinant PSD?

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question. A correlation matrix is a symmetric positive semi-definite matrix with 1s down ...
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### Can I factor a rational expression of the form…

Given two equations $\displaystyle P_1 = \frac{1-X^2}{1-X^2}$ and $\displaystyle P_2 = \frac{1-aX^2}{1-bX^2}$ I am told that there is a relationship between P1 ...
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### Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
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### Does encountering a zero pivot during Gaussian elimination imply that the matrix is singular?

I was reading a problem about Gaussian elimination and pivots of a matrix, say $A$. The question is: During the Gaussian elimination process without pivoting a zero pivot has been encountered. Is ...
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### Can we define component of a matrix which is orthogonal to another matrix?

Given two vectors $A$ and $B$ one can easily find component of $A$ along $B$ and component of $A$ perpendicular/orthogonal to $B$ and vice versa. This is possible as we can define dot product of two ...
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### Skew-Symmetric Parts of Stochastic Matrices

It's easy to see that the set $\{W - W^T : W \in \mathbb{R}^{n \times n}\}$ is precisely the set of real skew-symmetric matrices. This continues to be the case if we restrict to (entry-wise) non-...
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### Prove that the number: $z = \det(A+B) \det(\overline A-\overline B)$ is purely imaginary.

Problem: Let $A_{n\times n}$ and $B_{n\times n}$ be complex unitary matrices, where n is an odd number. Prove that the number: $$z=\det(A+B) \det(\overline A-\overline B)$$ is purely imaginary. My ...
How can I compute the rotation matrix which rotates an $n$-dimensional vector $\vec{A}$ around an $n-D$ vector $\vec{O}$, and maps it to a vector $\vec{B}$ (while $\vec{A}, \vec{B}, \vec{O}$ are known)...