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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
5 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
1
vote
0answers
17 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
1
vote
2answers
45 views

Why is this the eigenvector?

For the eigenvector how are they getting \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} when you have \begin{bmatrix} 0 & -1 & -1 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{bmatrix} ...
1
vote
1answer
20 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
0
votes
2answers
36 views

How do you solve this circular system of equations in $\mathbb{Z}_2$?

I'm trying to solve a system of equations in $\mathbb{Z}_2$ that look like this: \begin{align} x_1 + x_2 = p_1 \\ x_2 + x_3 = p_2 \\ x_3 + x_4 = p_3 \\ ... \\ x_n + x_1 = p_n \\ \end{align} I know ...
1
vote
1answer
12 views

What the limit of a matrix over time shows about the future

$x_k$ is the fraction of people who prefer cake to pie at year $k$. The remaining fraction $y_k=1-x_k$ prefer pie. At year $k+1$, $\frac{1}{5}$ of those who prefer cake change their mind. Also at year ...
1
vote
2answers
33 views

Matrix with eigenvalue that should equal 1.

I have the matrix: $$A = \begin{bmatrix}4 & -2 & 3\\0 & -1 & 3\\-1 & 2 & -2 \end{bmatrix}$$ and I need to find out if $\lambda = 1$ is an eigenvalue. So I solved the equation ...
0
votes
0answers
11 views

How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
1
vote
0answers
12 views

Uniqueness of pseudoinverse?

Recently, I read a line of reasoning as Since $A$ (a $3\times 3$ matrix) is of rank $2$, its pseudoinverse is not unique. May I ask if there is a quickie to show this?
1
vote
4answers
39 views

A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then :

A real $2 \times 2 $ matrix $M$ such that $$M^2 = \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$$ (a) exists for all $\epsilon > 0$. (b) does not exist for any ...
2
votes
1answer
23 views

How can I prove that any matrix A can be expressed as the sum of two Hermitian matrices , B and C, in the form A = B + iC?

The question is in the title really. Whether or not A must also be Hermitian is not clear to me. Sorry, I am not very good with proofs of this nature.
1
vote
0answers
12 views

Using Levi-civita symbol to determine axis and angle of rotation matrix

One of the questions on the course involves finding the angle and axis of this rotation matrix; $$R = \gamma\ \begin{pmatrix} 0 & -2 & 1\\ 2 & 0 & 0\\ -1 & 0 & 0 ...
1
vote
2answers
37 views

Finding a matrix from equation

we've got the following 4x4 Matrix $$\begin{pmatrix} 4 & -2 & 3 & 2\\ 3 & 5 & 1 & -4\\ -1 & 6 & -4 & -7\\ -2 & 0 & -2 & 4 \end{pmatrix}$$ and I need ...
-1
votes
1answer
20 views

Proving multilinearity of determinant [on hold]

As the title says, how we can prove multilinearity property of determinants: $$ \begin{vmatrix} p+q+r & x+y+z & u+v+w \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\\ ...
2
votes
1answer
33 views

If $A$ is positive definite then so is $A^k$

Prove that if $A$ is positive definite, then so are $A^2,A^3,\ldots$ and $A^{-1},A^{-2},\ldots$ I know how to show the inverse of positive definite is positive definite but I don't know how to expand ...
3
votes
3answers
28 views

Proving $AD_1A^{-1}=D_2$

I want to prove that if $A$ is a permutation matrix, and $D_1$ is diagonal, than $AD_1A^{-1}=D_2$ where $D_2$ is also a diagonal matrix. I have worked out that $A^{-1}=A^T$ and I can see that the ...
0
votes
0answers
26 views

centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
1
vote
1answer
26 views

Positive Semidefinite Matrices

Let $x=\left[ \begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{n}% \end{array}% \right] $ be a vector with $\sum x_{i}=1$ and $x_{i}>0$. Is there an easy way to prove that ...
0
votes
0answers
16 views

A 2D smoothing convolution filter

I'm trying to find the right form of a 2D filter that will do the following to a matrix after linear convolution: Let A = [ ? ? ?] [ ? ? ?] [ ? ? ?] and B = ...
1
vote
0answers
8 views

Inverse properties of $L_1$ normed matrices

Let's take the adjacence matrix $A$ of a graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. Let's take some $\alpha \epsilon [0,1)$. $(I-\alpha\bar{A})$ is strongly ...
0
votes
2answers
30 views

Understanding matrix property

I am reading about matrix property from here. On page 2 of pdf (equation 2.2), it says if $A$ is a matrix and $U$ a row-echelon form of $A$ then $$|A| = (-1)^r \alpha |U| ...
0
votes
1answer
24 views

Sum of orthogonal matrices

Consider the subspace $\mathbb{R}^m$ with usual inner product.Let $S_1$ and $S_2$ subspaces of $R^m$, $P_1\in M_m(\mathbb{R})$ the orthogonal projection matrix on $S_1$ and $P_2\in M_m(\mathbb{R})$ ...
2
votes
3answers
102 views

How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
-1
votes
0answers
32 views

Fill in the missing entries of matrix $Q$ to make it orthogonal [on hold]

I am given the following matrix $Q$: $Q=$ where $p1,p2,...,p8$ are unknowns. I need to make $Q$ into an orthogonal matrix. It occurs to me that $v1 =\{1,1,1,1\}$ and $v2=\{2,1,0,-3\}$, but I'm ...
3
votes
0answers
18 views

On the probability of singular matrices containing whole numbers

Today in class - my teacher was teaching determinants . He gave us problems to solve of various kinds , including various row - column operations and determinants properties. But one thing that ...
-4
votes
1answer
27 views

Orthogonality and inner product [on hold]

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
0
votes
0answers
15 views

Notation: column/row projection function for matrix-like objects

If we have a $n$-tuple $\mathscr x$ $$\mathscr x := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
0
votes
2answers
33 views

Nilpotent matrix similar to a matrix $[0,X]$ where $X$ is full column rank.

I am trying to prove that a nilpotent matrix $N$, which has a Jordan Form consisting only of blocks which are order 2 or greater, is always similar to a matrix $\begin{bmatrix}0 & X\end{bmatrix}$ ...
0
votes
1answer
24 views

Product of two multivariate Gaussian pdfs - normalization constant

https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ ...
3
votes
1answer
36 views

Deducing that the inverse of a permutation matrix is its transpose

I would like to verify that my proof below is sound. Let $A\in P$ where $P$ is the set of all permutation matrices (only one 1 in each row and column). Also, let $(A)_{ij}$ denote the entry of $A$ in ...
1
vote
1answer
21 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
-1
votes
1answer
15 views

Null space and Matrix equations

http://studyguide.pk/Past%20Papers/CIE/International%20A%20And%20AS%20Level/9231%20-%20Further%20Mathematics/9231_s03_qp_1.pdf I would like to know the method to answer question 8. I have been having ...
1
vote
1answer
56 views

What is the difference between $A^{-1}$ and $A^\Theta$?

Let $A$ be a square invertible matrix. Then $$A \cdot A^{-1} = I$$. Let $A^\Theta$ be the conjugate transpose matrix of $A$. Then $$A \cdot A^\Theta = I$$. Both on multiplication with $A$ gives ...
1
vote
0answers
24 views

Proving that a linear functional is matrix trace [duplicate]

Let $W=\operatorname{M}_{n\times n}(\mathbb{F})$ (square matrices $n\times n$ over $\mathbb{F}$), and $f\in W^*$. If $f(AB)=f(BA)$ for every $A,B\in W$ and $f(I)=n$ prove that ...
2
votes
0answers
20 views

Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
1
vote
0answers
11 views

eigenvector perturbation

In the proof of Theorem 1 of (http://ai.stanford.edu/~ang/papers/ijcai01-linkanalysis.pdf), the authors cite a theorem from Steward and Sun (Theorem V.2.8) which states that if $S$ is symmetric and ...
0
votes
0answers
16 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
1answer
22 views

Which geometric operations are encoded by symmetric, positive definite matrices?

Maybe it's because I'm German and used different terminology in the past, but somehow I don't really understand what is meant by the question in the title. Didn't change the wording, just copied it. ...
0
votes
1answer
51 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
0
votes
2answers
34 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
1
vote
3answers
29 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
1
vote
5answers
77 views

Product of any two arbitrary positive definite matrices is positive definite or NOT? [duplicate]

Suppose that , $A$ and $B$ are $n\times n$ positive definite matrices and > $I$ be $n\times n$ identity matrix. Then which of the followings are positive definite ? (i) $A+B$ (ii) $ABA$ ...
0
votes
1answer
24 views

Inconsistent Matrices

I'm teaching myself Linear Algebra and am not sure how to approach this problem: Let A be a 4×4 matrix, and let b and c be two vectors in R4. We are told that the system Ax = b is inconsistent. ...
4
votes
1answer
41 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
5
votes
4answers
346 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
0
votes
1answer
17 views

Finite differences matrix and integrals

Let $f:[a,b]\to \mathbb{R}$ a smooth function. Consider a partition $a=x_1<x_2<\ldots<x_n=b$. If we put $X=(f(x_1), f(x_2), \ldots, f(x_n))$, where $x_{i+1}-x_i=\Delta x$ then: $ (f'(x_1), ...
0
votes
1answer
43 views

Overdetermined linear system solutions proof

Let $A\in M_{m\times n}(R)$ with $m>n$. Consider that the only solution of the linear homogeneous system $Ax=0_{R^m}$ is the trivial solution $x=0_{R^n}$. Show that linear system $A^ty= b$ have ...
-8
votes
0answers
45 views

Which of these are true? And why? [on hold]

Which of these are true? And why are they true, please answer this too, if possible?
0
votes
1answer
11 views

Is there any shortcuts in getting an H-infinity norm of a matrix expression?

One of the past exam problems I was solving, has this in its official solution: Usually, to calculate the $H_{\infty}$ norm of any matrix expression $M$ I'd first calculate the eigenvalues of ...
0
votes
2answers
47 views

Propositions of elementary matrix

i'm trying to solve a question about elementary matrix. When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix. Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$; ...