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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-1
votes
0answers
16 views

How to solve an initial value problem consisting of a matrix?

I am doing differential equations problems and the problem states this: Solve the initial value problem \begin{align} X'(t) &= \begin{pmatrix} 1 & -2 & 2 \\ -2 & 1 & -2 \\ 2 ...
20
votes
9answers
11k views

Do all square matrices have eigenvectors?

I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation. And, so far I thought every matrix has eigenvectors. Please ...
0
votes
0answers
3 views

Bounding Product of Nonnegative Matrices

Let $A \in \mathbb{R}^{n \times m}_{\geq0}$, not necessarily square, have nonnegative elements, and let $x \in \mathbb{R}^m_{\geq 0}$ be nonnegative. Clearly $Ax$ is a nonnegative vector. Suppose in ...
7
votes
3answers
63 views

Showing $A+B$ is invertible?

Question number two of this released exam asks: Let $A$, $B$ be two $n \times n$ matrices with real elements such that $A^3 = B^5 = I_n$ and $AB = BA$. Prove that $A+B$ is invertible. I am not ...
0
votes
2answers
67 views

the minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$

The minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$.
1
vote
1answer
16 views

Determining similar matrices

I have this matrix $$A= \begin{bmatrix}1 &0& 2\\0&-1&-2\\2&-2&0\end{bmatrix}$$ I found the eigenvalues to be $0, 3, -3$ I am tasked with finding if $A$ is similar to a ...
0
votes
1answer
16 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
0
votes
3answers
41 views

What do I conclude if I found the eigenvalues of a matrix, then noticed that one of those eigenvalues resulted in a zero eigenvector?

By definition: $$Ax = \lambda x$$ for $x \neq 0$ I was using this to calculate the eigenvectors for $A$: $$A = \begin{bmatrix}2 & 0 & 0\\1 & 3 & 0\\ 2 & 3& 4\end{bmatrix}$$ ...
7
votes
1answer
81 views

Does $AB+BA=0$ imply $AB=BA=0$ when $A$ is real semidefinite matrix?

This is a general question that came to my mind while listening to a lecture(although its framing may make it look like a textbook question). Suppose that $A$ and $B$ be real matrices. $A$ is ...
0
votes
3answers
60 views

If $A$ and $B$ are arbitrary $n \times n$ matrices, prove that $(A^TB^TBA)$ is symmetric

My attempt: $(A^TB^TBA)^T$=$(A^T)^T(B^T)^TB^TA^T$=$(AB)B^TA^T$ $\ne$ $(A^TB^TBA)$ therefore $(A^TB^TBA)$ is not symmetric.
0
votes
0answers
14 views

Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
1
vote
1answer
27 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
0
votes
0answers
7 views

sum converge, matrix, norm

Let $A_j$ be a sequence in $\mathbb{C}^{n\times n}$. Show that $ \sum_{j=0}^\infty A_j$ converges if $ \sum_{j=0}^\infty ||A_j||$ does.($||A||= sup_{|x|=1} |Ax|$ with euclid norm) Hello, Be ...
1
vote
0answers
25 views

Why does Correlation Coefficient concern about the mean of the vector?

$$r = \frac {\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\sum_{i=1}^n(Xi-\bar X)^2} \sqrt{\sum_{i=1}^n(Y_i-\bar Y)^2}}$$ This is exactly the $\cos$ of degree of the angle between vector $X-\bar ...
0
votes
0answers
9 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in ...
3
votes
2answers
32 views
+50

A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & ...
0
votes
1answer
999 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
8
votes
3answers
3k views

Why are Vandermonde matrices invertible?

A Vandermonde-matrix is a matrix of this form: $$\begin{pmatrix} x_0^0 & \cdots & x_0^n \\ \vdots & \ddots & \vdots \\ x_n^0 & \cdots & x_n^n \end{pmatrix} \in ...
3
votes
4answers
297 views

Is it easier to determine that a matrix is singular than it is to determine nonsingular?

I came across this line "It is often easier to determine that a matrix is singular than it is to determine that a matrix is nonsingular. The facts below illustrate this. Fact 1.10. Let ...
0
votes
5answers
63 views

Why, if a matrix $Q$ is orthogonal, then $Q^T Q = I$?

I was looking at the definition of an orthogonal matrix, which is as follows: Square matrix $Q$ is orthogonal if its columns are pairwise orthonormal, i.e., $$Q^TQ = I$$ Hence also ...
0
votes
1answer
24 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
1
vote
0answers
17 views

Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Starting from the closet set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in ...
0
votes
0answers
22 views

Operator norm of a diagonal matrix

I want to prove that the operator norm of a diagonal matrix $D$ is less than or equal to its largest value. I've tried the following but I don't know if it is correct. ...
0
votes
2answers
35 views

Inverse of a square block matrix

I am trying to understand how to compute the inverse of a square block matrix defined as follow: $\begin{bmatrix}2{\bf I}&-{\bf X}\\{\bf X}'&{\bf 0}\end{bmatrix}$, where ${\bf I}$ is a ...
1
vote
2answers
63 views

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. [on hold]

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. I'm not sure how to do this. I know the result for $(I-A)^{-1}$, but that won't help me.
0
votes
0answers
39 views

Is there a variant of Euler's theorem for matrices?

Euler's theorem says that whenever $a$ and $p$ are coprime integers, then $a^{\phi(p)} \equiv 1 \mod p$ This is particularly useful for evaluating large powers of $a$: $a^n \ \%\ p = a^{n\ \%\ ...
0
votes
1answer
19 views

How come the determinant of a matrix have to be 0 to find the eigenvalue and vector?

I need help understanding why if the determinant of a matrix is 0 then there exists a matrix such that multiplying it by a vector get 0 and how this relates to eigenvectors and eigenvalues. For ...
2
votes
1answer
57 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
0
votes
1answer
18 views

Lower bound for the distance between matrices of different rank.

This is a follow up question to this: Norm of diference of matrices of different rank Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over ...
0
votes
0answers
16 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
0
votes
0answers
21 views

Identity involving pseudoinverses (Moore-Penrose) of symmetric matrices

Let A be a symmetric $m$ x $m$ matrix of rank r, and B a symmetric $m$ x $m$ matrix of rank $m - r$, such that $AB = 0$. Show that $A^+A+B^+B=I$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. ...
0
votes
1answer
21 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
1
vote
0answers
41 views

For matrices $A$,$B$ prove that $A+cB$ is not invertible.

Let $A$,$B$ $\in M_n(\mathbb{R})$ and $B$ is invertible, then prove that there exists a $c \in \mathbb{R}$ such that $A+cB$ is not invertible. My attempt: We need to show that $\det(A+cB)=0$. So ...
2
votes
2answers
17 views

Inverse of a matrix with uniform off diagonals

Suppose that we have an all positive matrix where the off diagonal elements are all identical. Can one calculate the inverse of the matrix analytically, or more efficiently than the general case? For ...
1
vote
0answers
18 views

Linear transformations and possible dimension mismatch

The problem: Let $L: R_4 \to R_3$ be defined by $$L([u_1, u_2 ,u_3 ,u_4]) = [u_1 ,(u_2+u_3), (u_3 + u_4)]$$ Let S and T be the natural bases for $R_4$ and $R_3$, respectively. Find the ...
1
vote
1answer
38 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
0
votes
3answers
39 views

What is the difference between a linearly independent set and a set that spans $\Bbb{R}^m$?

This is more of a conceptual question. Here's what I know about a linearly independent set of vectors: A set of vectors $\{v_1, ..., v_p\}$ is linearly independent if the equation $$x_1v_1 + x_2v_2 + ...
2
votes
1answer
34 views

Idempotent and nilpotent matrices are defined differently. Why?

We call $A$ idempotent if $A^2$ is $A$. But we call A nilpotent if $A^k$ is $0$ for some integer $k$. Why are not they defined uniformly like both with power 2 or both with power some integer $k$.
0
votes
1answer
22 views

Gradient Chain Rule: Applying Gradient in the case of a Series of Matrix operations (Neural Net Gradient Calculation)

I have the following situation: I need to calculate the gradient of the Error of a CNN a few layers deep by hand. Starting with the Error function, The $\operatorname{Error}[readoutX]= -\sum_i ...
3
votes
2answers
39 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
0
votes
1answer
32 views

Linear Regression without X? :

(Have been working in matrix algebra) Given model: $ y_i = a + e_i$ ( $y_i= α+ϵ_i$ ) That is $y$ subset $i$ and error term subset $i$ Where the expected value of each error term for each entry ...
-3
votes
2answers
26 views

An example of unitary matrix which is $3\times 3$ and complex

Please give me an example of unitary matrix which is $3\times 3$ and complex. If I get this example, i will finish my thesis.
0
votes
1answer
13 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...
0
votes
0answers
9 views

Solve a generalized eigenvalue problem in LDA

http://www.facweb.iitkgp.ernet.in/~sudeshna/courses/ML06/lda.pdf Page 6. I don't quite understand how that can be solved... I have tried following general one $$det(S^{-1}_{w}S_B-JI)=0$$ But I am ...
2
votes
0answers
23 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
3
votes
1answer
50 views
+100

How to prove this result about the interlacing of eigenvalues.

Let $A$ be a real symmetric matrix of order $n$ with eigenvalues $\mu_1\ge \mu_2,\ldots, \mu_{n}$. $B=\begin{bmatrix} x&x&x&x&x \end{bmatrix}$, where $x$ is non zero column vector in ...
-4
votes
1answer
34 views

Is it unitary matrix or not? [on hold]

$A = \begin{bmatrix} \frac{i}{3^{1/2}} & \frac{1+i}{3^{1/2}} & 0\\ \frac{-1}{2^{1/2}} & 0 & \frac{i}{2^{1/2}}\\ \frac{1-i}{3^{1/2}} & \frac{1}{3^{1/2}} & 0 \end{bmatrix}$ Is ...
0
votes
1answer
13 views

Hexagonal - number of cells

For $n = 2$; We have something like this: https://zapodaj.net/0cc6e3c190f32.png.html and number of calls is equal 7. But how designate for $n$ ? For $n = 3$; we have 19
0
votes
1answer
19 views

Columns of a matrix linearly independent and spans

Let $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n \in \Bbb R^n$ and let $P$ be the $n\times n$ matrix whose columns are $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n$ I'm wondering why the followings are ...
0
votes
0answers
4 views

partial order and equivalence relation question [on hold]

Let A = ℤ+ x ℤ+ and R be a relation on A (that is, R ⊆ A xA) defined as follows. (a,b) ~ (x,y) if and only if a + y = b + x. Is R a partial order? Is R an equivalence relation?