Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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2
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1answer
54 views

How to calculate a determinant of a 2x2 symmetry block matrix?

I'd like to calculate the determinant of the matrix: $$ \begin{pmatrix} -A & B^\star \\ -B & A^\star \\ \end{pmatrix} $$ $A$, $B$ are $L\times L$ complex ...
0
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1answer
22 views

Linear independence, if $ |a_{kk}| > \sum_{i=1, i \neq k}^{s} |a_{ik}| $

I found task written below, but I cannot prove it. Given is system of $s$ vectors ($a_i = (a_{i1}, a_{i2}, \dots a_{in})$ for $i = 1, \dots, s$), where $s \leq n$. Prove that, if for all $1 \leq ...
-1
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3answers
61 views

Bounds on determinant [closed]

I was just curious about the bounds on the determinant of a 3x3 matrix whose elements take values between 0 and 5. I believe this bound is around +-1040
2
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3answers
44 views

Determinant of an anti-diagonal block matrix

Is it true in general that if $A$ and $B$ are two $n \times n$ matrices, then the determinant of the anti-diagonal block matrix $$ J = \left[\begin{array}{cc} 0 & A \\ B& 0 ...
0
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0answers
35 views

How to calculate a determinant of a general 2x2 block skew matrix?

I would like to calculate the determinant of a $2\times2$ block skew matrix: $$ \begin{pmatrix} A & B^T \\ -B & D \\ \end{pmatrix} $$ with $A^T=-A$ and ...
1
vote
1answer
23 views

Limits, Determinants and Inversion of a matrix-valued function

Suppose I have a matrix-valued, continuous function $$A\colon [0,\infty) \to \mathbb R^{n\times n},\qquad h\mapsto A(h).$$ I know that for the limit $h\to 0$ the matrix is invertible: ...
7
votes
2answers
112 views

Problem involving trace and determinant of symmetric matrices

I've stumbled upon this exercise on a linear algebra book that asks me to determine all the ordered pairs $(a,b)$ of real numbers to which there exists an unique symmetric matrix $A\in R^{2\times 2}$ ...
3
votes
5answers
51 views

Prove that if P is idempotent a $I- \lambda P$ is invertible

Let $P\in K^{nxn},P^2=P$ and $\lambda \in K,\lambda \ne 1$. I need to prove that $I- \lambda P$ is invertible. I'm quite confused with this problem, because I know that if $P^2=P$ and $P\ne I$, then ...
0
votes
1answer
27 views

Gaussian integral for a vector and a function - how to evaluate

My problem concerns evaluation of a Gaussian integral. Let there be a real vector $\mathbf{v}$ and a matrix $\mathbf{A}$. I would like to know the result of the following integral: $$ ...
1
vote
1answer
47 views

Determinant of adjoint

I am trying to show that for $T$ over a complex inner product space we have $\det adj(T)=\overline{\det (T)}$. But I have seen this, which confirms this result over the reals: ...
1
vote
2answers
47 views

If $A$ and $B$ are $n$ x $n$ orthogonal matrices and $|A|+|B|=0$. Show that $|A+B|=0$ [duplicate]

If $A$ and $B$ are $n$ x $n$ orthogonal matrices and $|A|+|B|=0$. Show that $|A+B|=0$ Since $A$ and $B$ are orthogonal we know that their determinant is plus or minus $1$. I'm not sure how to show ...
4
votes
2answers
63 views

Find a complicated determinant

Find the determinant of the $m \times m$ matrix $K$ where $$K_{ij} = {1 \over 1 - x_ix_j} $$ for any values of $x_1,x_2,\dotsc, x_m$. My first thought is to make each component polynomial by scaling ...
0
votes
1answer
37 views

Proof that a particular matrix form is positive (semi-) definite

Let $$\mathbf{E} = \left[ \begin{array}{ccc} 1 & a_1 & b_1 \\ 1 & a_2 & b_2 \\ \vdots & \vdots & \vdots \\ 1 & a_n & b_n \end{array} \right] $$ where $0 < a_i,b_i ...
0
votes
1answer
26 views

Linear Algebra Proof - Columns of Matrix Linearly Independent & Determinant

How can I prove that if the columns of matrix A are linearly independent, then det(A) does NOT equal zero? This is a question on my exam review and I have no idea how to go about proving this. Any ...
0
votes
1answer
60 views

If $\lambda$ is an eigenvalue of $C$, prove $\frac{1}{\lambda}$ is an eigenvalue of $C^T$

$C$ is an orthogonal matrix. If $\lambda$ is an eigenvalue of $C$, prove $\frac{1}{\lambda}$ is an eigenvalue of $C^T$. I know $\lambda$ isn't zero because an orthogonal matrix has determinant $1$ ...
0
votes
0answers
16 views

what is the least value of modulus of x?

I did the 1st and 2nd part but I am having problem with the 3rd part. according to the given formula of magnitude i am getting |x|=square root of 15. which is not the answer (answers are given below). ...
0
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0answers
22 views

solution to recurrence relation involving terms of the form $\mathrm{tr}[(z^\dagger z)^k]$

While working on my research I encountered a recurrence relation which I am having trouble solving. The problem is the following: let $z$ be a $n\times n$ skew-symmetric (but not necessarily ...
1
vote
2answers
90 views

Matrix Determinant inequality.

I'm having some trouble in proving this inequality: $$\text{Det}\left(\frac{1}{n}\mathsf{A}^n\right) \leq \frac{1}{n-k}\text{Det}\left(k \mathsf{A}^{n-k}\right)$$ For every $k < n$, $k\neq 0$, ...
0
votes
0answers
34 views

Question about representing certain determinant in a general form.

So here is the question; it is to prove $$\det\begin{bmatrix}1&1&0&0&\cdots& 0&0\\0&1&1&0&\cdots& 0&0\\0&0&1&1&\cdots& ...
2
votes
2answers
96 views

Determinant of a $n\times n$ Matrix

Let $A = (a_{ij}) \in R^{n\times n}$. Find the determinant if: $$a_{ij}= |i-j|$$ So we have the symmetric matrix \begin{bmatrix} 0 & 1 & 2 & 3 & 4 & \dots & n-1 \\ 1 & 0 ...
1
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2answers
21 views

Prove the characteristic equation is of degree n

I'm attempting to prove the following theorem. Let $A \in M_{n \times n}(F)$ The characteristic polynomial of $A$ is a polynomial of degree $n$ with leading coefficient $(-1)^n$ The theorem itself ...
2
votes
1answer
36 views

How to find deteminant of the tridiagonal matrix

Given the matrix: $$ A = \begin{pmatrix} x & 1 & 0 & 0 & \cdots & 0 & 0 \\ n - 1 & x & 2 & 0 & \cdots & 0 & 0 \\ 0 & n - 2 & x & 3 & ...
0
votes
1answer
16 views

Matrix system solutions

I am doing the following exercise in preparation for an exam: I have solved (a) and (b) (a) $\lambda(2\lambda^2-6\lambda+4)$ (b) $\lambda = {0,1,2}$ However I don't know the answer to (c). If I ...
0
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0answers
10 views

Is this the correct way to interpret results from the principal minor test?

Am I correct in thinking that, given all the principal minors $A_{i}$ of the matrix associated with a quadratic form $Q(x,y)$, the Principal Minor Test states that $Q(x,y)$ is Positive Definite if ...
1
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0answers
11 views

Efficiently compute the Determinant of a Banded Matrix

So I've got a large (~ 2 million x 2 million) positive semi-definite, banded, square matrix that I need to find the determinant of. What is the correct way to efficiently compute the determinant of a ...
0
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0answers
23 views

Sarrus Rule proofed by Leibniz Formula

I want to proof the Sarrus Rule by using the Leibniz Formula. I am not quite sure about the permutations and how to get which ones I need to multiplicate. Thank you.
3
votes
1answer
36 views

Is the function determinant $A \rightarrow \det(A)$ a non-convex fuction?

Is the function $$ \det: A\in \mathbb{M}^{n \times n}(\mathbb{R}) \rightarrow \det (A)$$ a convex function? I think the answer is no, but I cannot prove it directly using the definition of convex ...
1
vote
1answer
29 views

Proof strategy about a property of triangular matrices

Is it by mathematical induction the best way to prove that the determinant of an upper (lower) triangular matrix is the product of the elements of the main diagonal? Actually, I am wondering about ...
0
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0answers
40 views

How to prove this determinant equation?

Given two matrices $A \in \text{Mat}_{n\times m}$ and $B \in \text{Mat}_{m\times n}$ for $m \geq n$, I need to prove this: $$\det(AB) = \sum\limits_I \det(A_I)\det(B_I),$$ where I passes (?) all ...
3
votes
0answers
20 views

generalizations of the linearly independence of column vectors in a Vandemonde matrix to higher dimensions

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandemonde matrix the maps $$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto ...
2
votes
1answer
45 views

Prove that, $det A$ is invertible $\iff$ $A$ has an inverse.

Let $\mathbb K$ be a commutative ring with unity. Let $det:\mathbb K^{n\times n}\to \mathbb K$ be determinant function. Prove that, $det A$ is invertible $\iff$ $A$ has an inverse. I proved ...
3
votes
2answers
87 views

Determinant tridiagonal matrix [duplicate]

Can anybody help me out with getting an expression of the values of $\lambda$ for a matrix $A$ for which $det(A-\lambda I)$ equals the determinant of a matrix with on the main diagonal $-\lambda$, on ...
1
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2answers
40 views

Eigenvalues of a certain tridiagonal matrix

Consider $A \in M_n(\mathbb R)$ defined by: $$A=\begin{bmatrix} a & -1 & 0 & \cdots & 0 \\ -1 & a & -1 &\cdots& 0\\ 0 & -1 & a & \cdots & 0\\ \vdots ...
1
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0answers
40 views

The determinant of skew-symmetric matrix of even order is a square as a polynomial in matrix elements

Help me, please, to solve this task: Prove that the determinant of skew-symmetric matrix of even order have a full square as a polynomial in matrix elements
1
vote
3answers
89 views

Prove that $\det( P^{-1} AP) =\det(A)$

Let $A$ be an $n × n$ matrix and let $P$ be an $n × n$ invertible matrix. Prove that $\det( P^{-1} AP) = \det(A)$ Pretty lost on this one, partially because I don't understand the relationship ...
1
vote
4answers
71 views

If A is a matrix satisfying $A^3 + 4A - 2I = 0$, explain why A is invertible…

If A is a matrix satisfying $A^3 + 4A - 2I = 0$, explain why A is invertible. -I understand that I can easily find a matrix that fits this condition and prove that its determinant is not zero, but ...
0
votes
0answers
19 views

Is this product of two product factorizations correct?

I am working on an induction proof and would like to know whether this product equality is true: $$\big (\prod_{i=2}^n (\lambda_i-\lambda_1) \prod_{n\ge i > j \ge 1}(\lambda_i - \lambda_j)\big )$$ ...
4
votes
0answers
34 views

(Group) homomorphism other than determinant?

(1) Let $\phi : GL(n,\mathbb{R}) \to \mathbb{R}\setminus\{0\}$ be a group homomorphism. I know that $\phi(A)=\mbox{det}(A)$ and $\phi(A)=1$ are two such examples. But, is there any other example of a ...
0
votes
2answers
54 views

How to show that the determinant of this matrix is in a nice product factorization,

Show that $$det \begin{bmatrix} 1 & 1 & \cdots &1 \\ \lambda_1 & \lambda_2 & \cdots &\lambda_n \\ \lambda^2_1 & \lambda^2_2 & \cdots ...
0
votes
2answers
48 views

$f(x) = \det(E+Ax)$; find a derivative of this function in $x_0 = 0$

How do I prove that $f'(x_0) =\text{tr}(A)$ where $x_0 = 0$ and $f(x) = \det(E+Ax)$ for some square matrix $A$? P.S. I'm pretty sure there exists a solution that doesn't use anything more ...
0
votes
1answer
34 views

Find an integer $c$ that satisfies the equation $|A|={1\over c}[(A)^3-3tr(A)+2tr(A^3)]$ where $A\in M_{3x3}(\mathbb{R})$.

Find an integer $c$ that satisfies the equation $|A|={1\over c}[tr(A)^3-3tr(A)tr(A^2)+2tr(A^3)]$ where $A\in M_{3x3}(\mathbb{R})$. I know that $c=6$ from wikipedia, but I don't know how to show it. ...
0
votes
2answers
32 views

Let $B = \{δ_1,…,δ_n\}$ be subset of $D(V)$ and assume there exists vector $0_V \ne v ∈ V$ satisfying $δ_i(v) = 0$. Show $B$ is linearly dependent

Let $n$ be a positive integer and let $V$ be a vector space of dimension $n$ over a field $F$. Let $B = \{δ_1,...,δ_n\}$ be a subset of $D(V)$ and assume that there exists a vector $0_V \ne v ∈ V$ ...
1
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2answers
38 views

simple showing inverse of matrix also upper triangular

I'm trying to show that A be a $ 3 x 3 $ upper triangular matrix with $det \ne 0 $. Show by explicit computation that $A^{-1}$ is also upper triangular. Simple showing is enough for me. $$A= ...
1
vote
1answer
51 views

A is a positive definite matrix iff its leading principal minors are positive

I am to prove that the a symmetric matrix $A$ is positive definite iff the leading principal minors of $A$ are positive. The forward implication is clear. Since the eigenvalues of a SPD matrix are ...
0
votes
0answers
14 views

Automorphisms of the exceptional Jordan algebra preserve the determinant and trace.

I am trying to show that the automorphism of the exceptional Jordan algebra $\mathbb{J}_3(\mathbb{O})$ preserve the determinant and trace. This algebra consists of $3x3$ Hermitian matrices over ...
0
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0answers
24 views

Jacobians and their Vector Coordinates

Define $f_2$ : $ℝ$ $\to$ $ℝ^2$ by putting $$f_2 (\theta)=(\cos(\theta),\sin(\theta)),$$ and for n $\ge3$ define $f_n: ℝ^{n-1}\toℝ^n$ inductively by setting $$f_n=(\theta_1, \theta_2, ...
4
votes
1answer
26 views

Matrices made of gluing $\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}$ blocks have a real determinant

Prove that matrices made entirely of blocks of the form $\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}$ have a real determinant. For example, we claim $$\Delta=\det ...
2
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0answers
44 views

Proof by Induction that relates to the Jacobian Determinant

Define $f_2$ : $ℝ$ $\to$ $ℝ^2$ by putting $$f_2 (\theta)=(\cos(\theta),\sin(\theta)),$$ and for n $\ge3$ define $f_n: ℝ^{n-1}\toℝ^n$ inductively by setting $$f_n=(\theta_1, ...
3
votes
1answer
66 views

Finding Maximum Determinant of a $6\times 6$ Matrix

What is the maximum possible determinant of a $6 \times 6$ matrix of $\pm1$? This is the maximum I reached: $$\begin{vmatrix} 1 & -1 & -1 & -1 & -1 & -1 \\ 1 & 1 & ...
0
votes
1answer
16 views

Changing order of determinant and limits

Given a Matrix X and another Matrix $M=e^X$ and $\det M =1$, I want to show that the Trace of X vanishes, e. g. $\mathrm{Tr} (X) =0$. I think that one can write it as following: \begin{equation} ...