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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2answers
61 views

What is this form of 'notation' called?

I was reading some of Max Tegmark's lecture materials and I found this little thing. Is there a name for it? Specifically, I am talking about $S_1$ R $S_2$ & $S_1$ R $S_2$ and the matrix. Is ...
0
votes
1answer
93 views

How do I answer this question during a test

Let v1 =\begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} , v2 = \begin{bmatrix} 0 &\\ 1\\ 0 \end{bmatrix} , H ={\begin{bmatrix} s &\\ s\\ ...
1
vote
1answer
54 views

Deriving with matrices, matrice equation

Can someone do this stepwise so that I am able to see what is going on. I am sure there is not many steps to take, but I am struggling to see the logic that goes from the left hand side to the right.! ...
1
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1answer
132 views

Matrix Algebra, Proof of some Trigonometric Identities

Please Refer to the image to see the problem. This was the easiest way to input the question as it has some difficult symbols to input from a keyboard. [Edit: Image with task replaced by $\LaTeX$:] ...
0
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2answers
64 views

If A is symmetric positive definite, is -A symmetric negative definite and why?

I ask this because I'm programming a function that does only take a symmetric positive definite matrix as input. But now I'm told give to the function the negation of such a matrix. That makes no ...
1
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1answer
95 views

What's wrong with this argument? (Diagonalizable matrices and spectral theorem)

Please consider all matrices to be in $M_n(\mathbb{R})$. Let $A$ be a positive definite symmetric matrix and $B$ a symmetric matrix. Then $A$ represent a positive definite scalar product $\Phi$ in an ...
1
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0answers
61 views

For $2 \times 2$ matrix, if $AB = -BA$, what properties would $A$ and $B$ have with regards to each other?

As title says, assume that $A$ and $B$ are two-by-two matrices. I want them to satisfy $AB = -BA$. What properties do these two matrices have with regards to each other? Sufficient properties would be ...
1
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4answers
652 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
13
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3answers
231 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
2
votes
7answers
443 views

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset?

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? A little unclear about this...
0
votes
1answer
113 views

Prove that an upper triangular matrix $A$, such that $A^*A = AA^*$, must be diagonal.

Let $A \in \mathbb{C}^{n \times n}$ be an upper triangular matrix that satisfies $A^{*}A=AA^{*}$. Prove that $A$ must be diagonal. My attempt is to partition $A$ as follows: $$ A = ...
1
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0answers
40 views

Similarity of a Companion Matrix to a Diagonal Matrix

Let $A$ be the companion matrix of a monic polynomial $f \in K[x]$ with $deg f=n$. Show that $A-xI_{n}$ is similar to a diagonal matrix with main diagonal $1,1,1, \dots , 1, f$. I tried my proof in ...
0
votes
1answer
28 views

Proving the equality: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$ where $P=(A-BD^{-1}C)^{-1}$

I am trying to prove the following equality that I need to use as an intermediate step to solve one of my problems. The equality is the following: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$, where ...
0
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2answers
62 views

If $\operatorname{rank}(AB)=n$, what are the $\operatorname{rank}(A)$ and $\operatorname{rank(B)}$?

$A$ and $B$ are $n\times n$ matrices. Any hints on how to solve this or where to find the answer are welcome
2
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3answers
155 views

Find $\exp(D)$ where $D = \begin{bmatrix}2& -1 \\ 1 & 2\end{bmatrix}. $

$$C = \begin{bmatrix}2& -1 \\ 0 & 2\end{bmatrix}\quad $$ I break it down into two matrices $$A = \begin{bmatrix}2& 0 \\ 0 & 2\end{bmatrix}\quad \text{and}\quad B =\begin{bmatrix}0 ...
1
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4answers
78 views

How to solve for matrix $A$ in $AB = I$

Given $B$ = $\begin{bmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1 \end{bmatrix}$ I know that $B$ is equal to inverse of $A$, how can I go backwards to solve for $A$ in $AB = ...
1
vote
1answer
256 views

Find square matrices $A, B$, such that $\exp(A + B) \neq \exp(A) \exp(B)$.

The question is as the Title stated: I picked a very easy example. However, I am afraid, I am missing something. The two matrices that I picked are $$A = \begin{bmatrix}1& 0 \\ 0 & ...
0
votes
2answers
1k views

Rotate a point around another point by an angle

In a two-dimensional matrix, how do I go about rotating a point A around a point B by Z degrees like in the problem below:
0
votes
1answer
349 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
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1answer
79 views

Dense symmetric positive definite matrix

How could one define a dense symmetric positive definite matrix (dimension $1000 \times 1000$) with uniformly distributed eigenvalues (with the smallest eigenvalue $1$ and the condition number $100$) ...
1
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0answers
29 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
2
votes
1answer
508 views

Invertibility, eigenvalues and singular values

I am confused about the relationship between the invertibility of a matrix and its eigenvalues. What do the eigenvalues of a matrix tell you about whether a matrix is invertible or not? Also, what ...
2
votes
1answer
30 views

Let $f\colon V \rightarrow W$ and $A$ be the matrix of $f$ for a certain bases, find $\dim(V), \dim(W), \dim(\text{null}(f))$ and $\dim(\text{im}(f))$

Let $f\colon V \rightarrow W$ and A be the matrix of $f$ for certain bases, find $\dim(V), \dim(W), \dim(\text{null}(f))$ and $\dim(\text{im}(f))$. $A$ is given by the following matrix: ...
2
votes
3answers
74 views

If the inner product of two vectors results in a positive definite matrix, does their commutative inner product result in a positive scalar?

We are given two row vectors $x, y \in \mathbb R^n $ such that the product $ x^T \cdot y $ yields a positive-definite $n\times n$ matrix. Does the inner product $ x \cdot y^T $ yield a positive ...
2
votes
1answer
95 views

Linear transformation $\mathbb R^3$ to $M^{2\times 2}$

Can you give me an hint on how to solve this? Consider the linear transformation $S: \mathbb R^3(x) \to M^{2 \times 2}(\mathbb R)$ defined by $$S(a_3x^3+a_2x^2+a_1x+a_0)=\begin{bmatrix} a_0 & ...
0
votes
2answers
95 views

Linear transformation and matrix basis

Consider the linear transformation $T:M_{2x2}(\mathbb{R}) \rightarrow M_{2x2}(\mathbb{R})$ defined by $T(A)= A^T$. Consider the basis $B$ of $M_{2x2}(\mathbb{R})$, defined by: ...
0
votes
1answer
57 views

Similar Matrices and Characteristic Polynomials

It is a well known fact that if two matrices $A$ and $B$ are similar, then they have the same characteristic polynomial i.e. det($xI-A$)=det($xI-B$). I think that the converse must also be true. As a ...
0
votes
3answers
3k views

Distance from Point on Plane to Origin

I have a practice question that is asking: Find the point on the plane 2x - 3y + z = 3 closest to the origin. What is the distance from that point to the origin? Here is my work so far (if you can ...
0
votes
2answers
441 views

Show that if $AB = BA$, then $\exp(A) \exp(B) = \exp(B) \exp(A)$ [duplicate]

Show that if $AB = BA$, then $$\exp(A) \exp(B) = \exp(B) \exp(A)$$ here is what i got so far: $$\begin{align}\exp(A) \exp(B) &= (I + A + {\dfrac{1}{2}} A^2 + {\dfrac{1}{6}}A^3 + \ldots)(I + B + ...
7
votes
2answers
171 views

trace , determinant and which of the following are true(NBHM-$2014$)

Let $A \in M_2(\mathbb R)$ be a matrix which is not a diagonal matrix . Which of the following statements are true?? a. If $tr(A)=-1$ and $detA=1$, then $A^3=I$. b. If $A^3=I$, then ...
0
votes
1answer
104 views

Find a symmetric matrix for f(x)

Consider the function $f(x,y,z)=2y^{2}+2xy+2xz+2yz$, Find a symmetric matrix $A$ such that $f(x,y,z)$ can be written in the form $(Ax)x=(Ax)^{T}x$, where $x^T = [x y z]$.
0
votes
3answers
70 views

Calculate determinant of $n \times n$ depending on n

My task is to figure out determinant of following matrix depending on $n$. I want to solve it without altering the rows! $$ A^{n,n} = \begin{vmatrix} 0 & & ... & 0 & -1\\ ...
1
vote
1answer
151 views

Geometrically Describing a Subspace

I have a practice question here and it is asking to geometrically describe this subspace in $R^3$ It is asking if it is a point, a line, or a plane or all of three-space. Here it is: Span {(1,-2, 1) ...
0
votes
2answers
275 views

Given the degrees to rotate around axis, how do you come up with rotation matrix?

Given angles (in degrees) to rotate around, $x$-, $y$-, $z$-axis how does one come up with the rotation matrix? For example if you have a point $p$ represented by a vector, how do you rotate it by ...
0
votes
1answer
58 views

A little help on geometric description of $\Bbb R^2$ in linear algebra

I just started studying vectors in linear algebra, and I didn't understand the idea of the geometric description of a vector. Why do we treat the vector entries as coordinates? As far as I ...
1
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1answer
61 views

condition number of product from 2 s.p.d matricies

In my numeric script the condition number for a invertable Matrix $A$ is definied as: $\mathcal{k}(A)= ||A^{-1} || ||A|| $ with euclidan norm. If $A$ is symmetric than it holds $\mathcal{k}(A)= ...
2
votes
0answers
48 views

linear algebra formulation help

I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by: $$ BE = ...
1
vote
1answer
52 views

How to find charpoly from eigenvalues and CH to prove an equation

For an uknown 3x3 matrix $A$ we know that $\operatorname{tr} A = 0$, $\det(A) = 1/4$ and we also know that two eigenvalues are the same. Proove that $4A^3 = -3A - I$. Problem says to use Vieta to find ...
0
votes
1answer
39 views

REF(A + B) = REF(A) + REF(B) [Strang P130 3.3.5]

Describe all $m$ by $n$ matrices $A$ and $B$ such that $ref(A) + ref(B) = ref(A + B)$. Is it true that $ref(A) = A$ and $ref(B) = B$? Does $ref(A - B) = rref(A - B)$? Here, ref = Row Echelon ...
0
votes
1answer
37 views

Sumation convention, proving matrix identity

On the lecture notes I found online there is the following proof that I don't understand. Let A, B be two matrices. $$(AB)^T=B^TA^T$$ I know this identity well, but I don't get this proof. Here it ...
7
votes
1answer
206 views

Maximum number of different diagonals obtained by column permutations

Consider a n x n matrix with entries being only '0' and '1'. For example: $\left( \begin{array}{ccc} 1 & 0 & 1\\1 & 1 & 0\\0&0&1\end{array}\right)$ We then consider all ...
4
votes
4answers
354 views

Quick question: matrix with norm equal to spectral radius

For $A\in \mathcal{M}_n(\mathbb{C})$, define: the spectral radius $$ \rho(A)=max\{|\lambda|:\lambda \mbox{ is an eigenvalue of } A\} $$ and the norm $$ \|A\|=max_{|x|=1}|A(x)| $$ where |.| is ...
2
votes
2answers
496 views

Are all rank-1 positive semidefinite matrices the result of an outer product $vv^T$?

If $v$ is a column vector in $\mathbb{R}^n$, then it is obvious that $V = vv^T$ is a rank-1 positive semidefinite matrix. I am wondering if the converse is true: if $V$ is a square real rank-1 ...
12
votes
3answers
752 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
3
votes
1answer
390 views

Trick to diagonalize symmetric matrices?

I will write an exam on Quantum Mechanics soon. I was wondering whether there is any smart and fast way to determine the eigenvalues/eigenvectors of a symmetric 3x3 matrix other than by calculating ...
0
votes
1answer
201 views

An example of matrix with spectral radius < 1

I am trying to run some tests on Jacobie iterative method for solving linear systems. However, I have a problem with finding such matrix $A$, which: isn't diagonally dominant when we take two ...
0
votes
1answer
78 views

Showing similarity of a companion matrix and a diagonal matrix

If $A$ is the companion matrix of a monic polynomial $f\in K[x]$, with deg $f=n$, show explicitly that $A-xI_n$ is similar to a diagonal matrix with main diagonal $1_K,1_K,\cdots,1_K,f$. Suppose ...
0
votes
2answers
40 views

Conceptual query for finding eigen values during change of basis

Consider an n x n matrix. Suppose i wish to find the eigen values of this matrix. Now, we know that row transformation is equivalent to a change of basis in the vector space. But, we also know that ...
0
votes
0answers
39 views

Find out the smallest disk like ($|z-1| < r$ ) in the complex plane containing the eigenvalues of the given matrix

Consider the given matrix $$\left[\begin{matrix}1& -2& 3& -2 \\1& 1& 0& 3\\-1& 1& 1& -1\\0& -3& 1& 1&\end{matrix}\right]$$ Find out the smallest ...
0
votes
1answer
328 views

Decomposition of idempotent and symmetric matrix

In Patterson and Thompson, 1971 I find the following claim which I cannot prove myself, nor have I been able to find a proof: "As $S$ is idempotent and symmetric [and of size $n\times n$ and with ...