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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4
votes
3answers
59 views

Matrix-by-matrix derivative formula

I need to derive $\frac{\delta(X^{T}MX)}{\delta X}$, where $X$ and $M$ are $n \times n$ matrices. I know that $\frac{\delta(AXB)}{\delta X}=B^{T} \otimes A$ but am having a hard time deriving what I ...
-2
votes
2answers
109 views

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 ...
2
votes
1answer
37 views

Determinant of a matrix and chech whether it is non negative definite or not

Let $V = \{ f : [0,1] \to \mathbb R | f$ is a polynomial of degree less than or equal to n $\}$. Let $f_j(x) = x^j$ for $0\leq j \leq n$ and let $A$ be the $(n+1) \times (n+1)$ given by $a_{ij} = \...
1
vote
1answer
46 views

derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the ...
5
votes
2answers
108 views

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
0
votes
0answers
20 views

Matrices in $F_n^m$ proof

Let F be a field and X, Y two F-linear subspaces with $dim_FX=n\in \mathbb{N}$ and $dim_FY=m\in \mathbb{N}$. To show: a)There exists a matrix $A\in F_n^m$ with $def(A)=0 \iff n\leq m$ b)There ...
0
votes
0answers
18 views

Making a specific matrix to be positive definite by pre- and post-multiplying (solving Lyapunov equation analytically)

I have a matrix in this form: \begin{equation} A=\left[ \begin{array}{ccc} -A_{11} && A_{12} && A_{13} \\ -A_{12}^\top && 0_{1 \times 1} && A_{23}\\ -A_{13}^\top &&...
1
vote
2answers
67 views

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 ...
1
vote
2answers
51 views

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$ Basically if I have found the orthonormal basis for the span of S can I use that to find the dimension ...
7
votes
2answers
93 views

If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$

Since the gcd of the integers $a_1,\ldots, a_n$ is $1$, there exists weights $x_i \in \mathbb{Z}$ such that $a_1x_1+\cdots+ a_nx_n=1$. My two ideas are (a) to brute force construct an $n\times n$ ...
0
votes
1answer
22 views

Property of an invertible matrix that row reduced form is identity matrix

If a the row reduced form of a $n \times n$ matrix is the equivalent $n \times n$ identity matrix. Is the $n \times n$ matrix always invertible? Furthermore if the row reduced form is not the ...
1
vote
0answers
45 views

I am confused about this notation for matrix representation.

I am confused about this notation from the image below. How can you represent something like (1,2),1 as a 2D matrix? For S1 = (0,1,2) and S2 = (0,1) I would expect two matrices like: [(1,0),(1,0),...
1
vote
1answer
36 views

Existence of matrices with non-zero principal minors

The problem sounds very simple but I have yet to come to an answer. Prove or disprove: For all $n$ there exists a matrix $A \in \mathbb{R}^{n \times n}$ with $\det(A) = 0$ such that all first ...
3
votes
1answer
41 views

$(AB=BA\wedge A^*Bx=0)\implies BA^*x=0$?

Let $X^*$ mean the conjugate transpose of matrix $X.$ I am given two matrices $A,B$ and a vector $x$ such that $AB=BA$ and $A^*Bx=0.$ Does $BA^*x=0$ then? It may look out of context, but such ...
1
vote
1answer
43 views

If $A$ and $B$ have the same degree of nilpotence, do they have the same rank?

Let $A, B$ be nilpotent $n\times n$ matrices over the field $K$. Is the following correct? If $A$ and $B$ has the same degree of nilpotency, then $\operatorname{rank} A = \operatorname{rank} B $
1
vote
1answer
53 views

Proof about isometries

i'm trying to prove this statements, but I don't find a starting point. Did someone have an idea how to prove this? Thanks in advance. Be $V=R^n$ furnished with the standard inner product and the ...
4
votes
5answers
388 views

Determine matrix of linear map

Linear map is given through: $\phi\begin{pmatrix} 3 \\ -2 \end{pmatrix} =\begin{pmatrix} -3 \\ -14 \end{pmatrix} $ $\phi\begin{pmatrix} 3 \\ 0 \end{pmatrix} =\begin{pmatrix} -9 \\ -6 \end{pmatrix}$ ...
0
votes
1answer
50 views

Prove the orthogonal matrix with determinant 1 is a rotation

Let's define "preserve orientation" in the following way (I am not sure it is right, pls point out if there is something wrong): For a linear transformation, we only need to check non-parallel ...
1
vote
3answers
56 views

Matrix with orthonormal base [closed]

I have the two following given vectors: $\vec{v_{1} }=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ $\vec{v_{2} }=\begin{pmatrix} 3 \\ 0 \\ -3 \end{pmatrix} $ I have to calculate matrix $B$ so that ...
0
votes
2answers
76 views

If $X$ is zero matrix,what is $e^X$?

Let $X$ be an n×n real or complex matrix. The exponential of $X$, denoted by $e^X$, is the n×n matrix given by the power series $e^X =\sum_{k=0}^{\infty} X^k/k!$ where $X^{0}$ is defined to be the ...
2
votes
3answers
170 views

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 ...
1
vote
3answers
33 views

Properties of RREF 3x3 matrix is the identity

The row reduced echelon form of a 3 × 3 matrix A is the identity. State whether each of the following is true or false. You do not need to explain your answers. (a) A has an inverse. (b) The columns ...
0
votes
0answers
41 views

Incorrect answer - Simultaneous Differential Equations

The questions states solve for y such that $$y' = \begin{bmatrix} -4 & 2 & 1 \\ 1 & -3 & 1 \\ 3 & -3 & -2 \\ \end{bmatrix}y , y(0)= c = \begin{bmatrix} 1\\5\\3 \end{...
0
votes
0answers
22 views

Relationship between matrix norms

Working in real space. Is spectral norm of a symmetric positive definite matrix greater than or equal to operator norm? Can you provide some inequalities between other norms like schatten norm, ...
0
votes
2answers
28 views

Proving $(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$ , $P$ idempotent matrix.

Given that a matrix $P$ is idempotent how to prove the following relation: $$(I -cP)^{-1} = I+ \left(\frac{c}{1-c}\right)P$$ $c$ is any real constant.
1
vote
1answer
31 views

Transformation matrices in a basis

Let $F=\mathbb{R}$,$X=\mathbb{R^3}$ and $Y=\mathbb{R^2}$. Further $B_X$ and $B_Y$ are given by: $B_X:=\{(1,0,0),(1,0,-1),(1,-1,-1)\}$ $B_Y:=\{(1,0),(1,-1)\}$ Let $f:X\rightarrow X$ and [...] be ...
1
vote
1answer
47 views

Computing the standard matrix of the linear transformation

Can you please explain this question to me? Suppose that $w = [1,2,3]^T$ and $L: \mathbb{R}^3\to \mathbb{R}^3$ is defined by $L(x) =\text{Proj}_w(x)$ (projection of $x$ onto $w$). Compute the ...
2
votes
1answer
44 views

What exactly does a rotation preserve?

I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a ...
-1
votes
1answer
19 views

Find the value of n , using eigenvector

I am unable to think how shall I proceed. I have to find value of n given a 2×2 matrix and an eigenvector. Can somebody help me out.
2
votes
2answers
56 views

For an orthogonal matrix $Q$, why does $QQ^T = I$?

In my linear algebra text (Strang), an orthogonal matrix is defined to be a square matrix whose columns are orthonormal. In other words, an orthogonal matrix is a matrix $Q = [q_1 \cdots q_n]$ where ...
1
vote
2answers
31 views

matrix calculation

Let $p= \begin{pmatrix} x & y \\ z & v \end{pmatrix}\in M_2(\mathbb{C})$ such that $p^2=\overline{p}^t=p$ and rank(p)=1. Why is $p=\begin{pmatrix} t & l\sqrt{t(1-t)} \\ \overline{l}\sqrt{...
1
vote
1answer
28 views

Exact same solutions implies same row-reduced echelon form?

In Hoffman and Kunze they have two exercises where they ask to show that if two homogeneous linear systems have the exact same solutions then they have the same row-reduced echelon form. They first ...
1
vote
1answer
37 views

Upper and lower bounds log determinant

I found an inequality in Wikipedia that i want to know how to prove it. For a positive definite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant. $...
0
votes
1answer
16 views

Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
2
votes
1answer
37 views

How to prove that $A$ is positive semi-definite if the symmetric minors are non-negative?

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix such that all its symmetric minors are non-negative (i.e. for $B=\left(a_{l_il_j}\right)_{1≤i,j≤k}$ with $1≤l_1<...<l_k≤n$ we have $\det(B)≥...
0
votes
0answers
38 views

At which points is function invertible?

Determine on which points is mapping local invertible? $f: \mathbb R ^{2}\mapsto \mathbb R ^{2}$ $\left(x,y\right)\mapsto\left(x^{2}-4y^{2}+x, -xy+3y \right) $ I calculated Jacobian matrix and ...
1
vote
1answer
47 views

How to verify that a matrix is a rotation matrix in Matlab?

Using Matlab, I want to know if $$A=\begin{pmatrix} \cos(x) & \sin(x)\\ -\sin(x) & \cos(x) \end{pmatrix}$$ is a rotation matrix. Hence, $$\begin{pmatrix} \cos(x) & \sin(x)\\ -\sin(x) &...
3
votes
0answers
50 views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
1
vote
1answer
38 views

Operator norm and eigenvalue inequality

Can I say that $\|A\| < s$ where $A \in \mathbb{R}^{3 \times 3}$ is a symmetric, positive definite matrix and $s$ is the maximum eigenvalue of $A$. Here the norm used is operator norm.
2
votes
1answer
42 views

What is $\mbox{Tr}^2(A)-\mbox{Tr}(A^2)$ in terms of the eigenvalues of $A$?

I am looking for a way to relate the terms of the characteristic polynomial of a $3 \times 3$ matrix to its eigenvalues. The definition I start with (taken from Wolfram MathWorld) is $\\P_{3}(A)=x^{...
1
vote
1answer
38 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
1
vote
2answers
102 views

$A^{n+1}=0\Rightarrow A^n=0$

A real $n\times n$-matrix $A$ satisfying $A^{n+1}=0$ must necessarily satisfy $A^n=0$. One way to see this is by looking at the Jordan Normal Form of $A$, another is by an argument involving the ...
0
votes
0answers
21 views

Can I do Gaussian elimination with a rectangular matrix

I'm writing some linear algebra scripts to understand the stuff I'm reading. I was wondering if I can only use square matrices as input for Gaussian Elimination, my guess is yes because permutation ...
0
votes
1answer
15 views

Significance of a capital R outside of brackets containing a matrix expression

I was trying to understand what a positive definite matrix is while reading a reinforcement learning paper today, and I came across this page: http://mathworld.wolfram.com/PositiveDefiniteMatrix.html ...
1
vote
1answer
62 views

Is this a correct way of proving that if $A^3-3A + I = 0$ then $A^{-1}=3I-A$?

$A^3-3A+I = 0$, I multiply both sides by $A^{-1}$ $$(A^3-3A+I)*A^{-1} = 0$$ $$A^3*A^{-1} - 3A*A^{-1} + I*A^{-1} = 0$$ $$A^3*A^{-1} - 3I + A^{-1} = 0$$ $A(A^2*A^{-1})-3I+A^{-1} = 0$ Here I ...
3
votes
1answer
37 views

additivity of rank

we know that for all $A,B\in M_n(\mathbb{C})$ : $$ rank (A+B)\leq rank(A)+rank (B) $$ see here for a simple proof, but for which condition on the coefficients of $A$ and $B$ we can obtain a perfect ...
4
votes
5answers
109 views

Calculating the matrix $M^{2006}$

Say you have the matrix $M$: $$\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$$ How do you find $M^{2006}$? My thinking was that you ...
2
votes
1answer
34 views

Find projection matrix using partitioned matrices

If X is a ($n$, $p+1$) design matrix, partition $X$ to be $X$=[$J$ $X$*] where $J$ is a ($n$,$1$) vector of all $1$'s, and $X$* is a ($n$,$p$) matrix. Let $H_X$ be a projection matrix, where $H_X$ =...
1
vote
1answer
14 views

Linear Functional - Basis change

Having a trouble understanding what you have to do in case you have a "simple" linear functional. The basis change and stuff for Quadratic and Bilinear forms are everywhere in every book, but there is ...
1
vote
1answer
25 views

3D Matrix Transformation

This is a question that stems from a programming problem I am having, but I do not understand the math behind it. So forgive me if there is some Stack Overflow bleed in my question. I have an array. ...