# 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), ...

39 views

15k views

### Understanding rotation matrices

How does ${\sqrt 2 \over 2} = \cos (45^\circ)$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function represent? ...
1k views

### How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 = ...
269 views

### Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is not diagonal or triangular. Is there a term for such matrices, and have they been researched?
28 views

### If both of $A,A^{-1}$ have entries from non negative integers then can we say $A$ is a permutation matrix?

I've shown if both of $A,A^{-1}$ (assuming $A$ to be invertible) are $n\times n$ matrices with entries from natural numbers then both of them have to be permutation matrices. Now my question is if ...
773 views

### Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$P^k = I$$ Find a 5 by 5 permutation $$P$$ so that the smallest power to equal I is $$P^6 = I$$ (This is a challenge question, Combine a 2 ...
52 views
+50

22 views

### Zeros in pivot position

When zero appears in a pivot position, $$A = LU$$ is not possible. What do we have to do here to make A=LU possible then? Do we have to find a specific P (permutation matrix) for A and continue ...
464 views

Let's assume that we want to find a rotation matrix which added to a given rotation matrix gives also a rotation matrix. I would name such matrix a rotation additive matrix for a given rotation ...
21 views

645 views

### Lower bound for the trace of product of two symmetric matrices

i am stuck on finding a lower bound of $tr(XY)$ of two symmetric matrices in $M_{n}(\mathbb{R})$. I know that it holds $tr(XY)=tr(YX)$ and thus $tr(XY-YX)=0$ and i can remember, that XY-YX is also ...
11k views

### Determine a basis for the solution set of the homogeneous system

Determine a basis for the solution set of the homogeneous system: \begin{align*} x_1 +x_2 +x_3 &=0\\ 3x_1+3x_2+x_3 &=0\\ 4x_1+4x_2+2x_3&=0 \end{align*} Then the augmented ...
68 views
+50

### Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
16 views

### Improper rotation matrx in $2D$

The following is the related problem: Improper Rotations in Even Dimensions I want the simpler explanation. An improper rotation is rotation, followed by reflection in the plane perpendicular ...
19 views

### Converting Fractional Coordinates to Cartesian

I'm confused about what I am reading online - different sites tell me different answers. Lets say I have a point pair in fractional coordinates, [xf,yf,zf]. I know that to convert them to their ...
70 views

Let $A$ be a square matrix such that all its eigenvalues are less than or equals 1 in absolute value. If A is not nilpotent, then prove that $$\text{There exists an } i_0 \text{ such that rank }(A^{... 2answers 51 views ### How to calculate matrix rotation Given the following rotation matrix$$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right] ...
I tried to solve the following problem and I'd like some feedback on my solution: Let $n$ be a positive integer and let $V$ be $P_n(\Bbb R)$the space of all polynomials functions over the field of ...