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.
6
votes
1answer
164 views
Do multiplicative maps of matrices factor through determinants?
Given a map $f:M_n(k)\to k$ (with $k$ some field) such that $f(AB)=f(A)f(B)$ for all matrices $A$ and $B$, is it necessarily the case that $f$ factors through the determinant, i.e. does there exist a ...
6
votes
1answer
178 views
Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$
The following is a historical question, but first some background:
Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
6
votes
0answers
111 views
Determining sign(det(A)) for nearly-singular matrix A
Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
5
votes
5answers
344 views
Is a square matrix whose diagonal and antidiagonal elements are all zero always singular?
Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices?
When $n=1,2,3,4$, the matrix is ...
5
votes
4answers
183 views
Linear Algebra: different determinant answers
I'm having a problem verifying my answer to this question:
Solve for x:
$$\left| \begin{array}{cc}
x+3 & 2 \\
1 & x+2 \end{array} \right| = 0$$
I get:
$(x+3)(x+2)-2=0$
$(x+3)(x+2)=2$
...
5
votes
2answers
153 views
Determinant called Grammian
Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if
$$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
5
votes
3answers
212 views
Determinant of a linear transformation defining matrix transpose
So if I define a linear transformation $ T: M_{n\times n}(R) \rightarrow M_{n\times n}(R) $ and $ T(A)=A^t $ what would be its determinant?
5
votes
4answers
210 views
Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$
Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$.
Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
5
votes
3answers
140 views
Formal proof of $\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i)$
I'm looking for a formal proof for:
$$\det(I + tA) = \prod\limits_{i=1}^n (1 + t\lambda_i).$$
I'm very new to matrix theory therefore please forgive me if you find this elementary. Your help in this ...
5
votes
2answers
456 views
where did determinant come from? [duplicate]
Possible Duplicate:
What's an intuitive way to think about the determinant?
I just learned the basics of matrices. Then I came across the magical formula
$$\det(AB)=\det(A)\det(B)$$
I ...
5
votes
3answers
68 views
How to show that $\det(A+I)\ne 0$
How to show that for any skew symmetric real matrix $A$, we have $\det(A+I)\ne 0?$
Where to begin? I'm looking for some clue only.
5
votes
3answers
96 views
Special orthogonal matrices have orthogonal square roots
Let $A$ be an orthogonal matrix with $\det (A)=1$. Show that there exists an orthogonal matrix $B$ such that $B^2=A$.
Thank you very much.
5
votes
3answers
195 views
computing determinant of a matrix
let $A$ be an $n\times n$ matrix with entries $a_{ij}$ such that
$a_{ij}=2$ if $i=j$.
$a_{ij}=1$ if $|i-j|=2$
and $a_{ij}=0$ otherwise.
compute the determinant of $A$.
using the famous formula ...
5
votes
3answers
329 views
Determinants of block matrices
Let $A,B \in \mathbb{R}^{n,n}$.
Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$.
Show that $\det(C) \in \mathbb{R}$ ...
5
votes
5answers
130 views
How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$
Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix
$$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$
How to prove that ...
5
votes
1answer
91 views
Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?
In a paper (corollary 1, p.14) the following identity is used:
Let g be a unitary matrix. Then:
$$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$
Now my question is why this ...
5
votes
1answer
147 views
What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?
If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$
I used ...
5
votes
2answers
338 views
Is the determinant of a zero divisor zero?
Suppose that $A$ is a zero divisor in the ring of $(n\times n)$-matrices over the ring $R$.
Is $\det(A) =0$ if $R$ is a field?
Is $\det(A) =0$ if $R$ is an integral domain?
It's not necessarily ...
5
votes
1answer
88 views
Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$
I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates ...
5
votes
1answer
114 views
Determinant of the transpose via exterior products
Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant:
If $t:U \to U$ is a linear operator and $\dim(U)=n$ then ...
5
votes
1answer
83 views
How to prove $\det(e^{\lambda_ix_j})\not=0$ where $\lambda_i\not=\lambda_j$ and $x_i\not=x_j$ if $i\not=j$
In try to figure out the exercise:
Let
$$f(x)=\sum_{k=1}^{n}c_ke^{\lambda_kx}$$where $\lambda_i \not=\lambda_j,i\not=j$,and $c_1^2+c_2^2+\dots+c_n^2\not=0$, then the number of $f(x)$'s roots is ...
5
votes
0answers
116 views
Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.
(this is an improved version of What about other symmetric functions of the eigenvalues? )
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
49 views
What about other symmetric functions of the eigenvalues? [duplicate]
Possible Duplicate:
Identities for other coefficients of the characteristic polynomial
Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
75 views
Determinant expression for the power sum
Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression:
$$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
4
votes
3answers
436 views
Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$
Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$.
My textbook only provides two simple examples.
Really have ...
4
votes
5answers
480 views
Find the determinant of $I+A$
Let $A$ be a $2\times2$ matrix with real entries such that $A^2=0$.Find the determinant of $I+A$ where $I$ denotes the identity matrix. I proceed in this way :Note that $(I+A)A=A+A^2 \Longrightarrow ...
4
votes
4answers
245 views
How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
4
votes
2answers
138 views
Is the determinant of a matrix lower when all its elements are lower?
Problem
Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix:
$$
A = \begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3}\\
a_{2,1} ...
4
votes
4answers
141 views
Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
This is a statement in linear algebra that I can't seem to understand the proof behind.
For a square matrix $A$, why is:
$$\mathrm{adj}(A)A = \det(A) \cdot I$$
Any explanation would be greatly ...
4
votes
4answers
245 views
Determinant of a special $0$-$1$ matrix
I have a matrix which is of odd order and has exactly two ones in each row and column. The rest of the entries in each row/column are all zero. What will be the determinant of this matrix?
I believe ...
4
votes
2answers
155 views
Why this determinant is conformally invariant?
While I was reading a paper about random analytic function I found a statement that I was not able to prove and after try brute force and search for some references I decided to ask for a help here. ...
4
votes
2answers
51 views
$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$
I found the following problem interesting but do not know how to tackle it.
If $A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$
Can ...
4
votes
2answers
340 views
Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant
If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?
I remember seeing such a ...
4
votes
3answers
321 views
Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?
I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
4
votes
1answer
166 views
Why is the determinant invariant under row and column operations?
I know that we may add any row to any other in a determinant and its value remains the same.
This is clear enough since elementary matrices corresponding to row and column operations have determinant ...
4
votes
2answers
78 views
How can I prove $\det(\overline M)=\overline{\det(M)}$?
Of course $\overline M$ is the complex conjugate of an $n\times n$ matrix $M$.
Someone gave me advice to use the definition of determinant, then it means I have to use cofactor expasion here?
4
votes
2answers
194 views
Interesting Determinant
Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$.
Define \begin{equation*}
A=%
\begin{bmatrix}
0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & ...
4
votes
2answers
104 views
Math hack for solving system of equations
Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
4
votes
2answers
522 views
The determinant of block triangular matrix as product of determinants of diagonal blocks
I am given the following partitioned - upper-triangular matrix:
$$
\begin{bmatrix}
A_1 &* &* &* &* &* \\
0& A_2 &* &* &* &* \\
.& 0& ...
4
votes
1answer
92 views
Simpler expression for a certain determinant.
A question in elementary linear algebra, while considering the Cayley-Menger Determinant:
Given an $n\times n$ matrix $M$, consider $$\tilde{M}=\begin{pmatrix} M & (1,1,\cdots, 1)^\top \\ ...
4
votes
1answer
69 views
When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?
What conditions, if they're even necessary, must be placed on $\star$ so that the matrix
$$ \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, ...
4
votes
4answers
327 views
Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues.
If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
4
votes
1answer
1k views
Determinant of a polynomial matrix
A matrix determinant (naively) can be computed in $O(n!)$ steps, or with a proper LU decomposition $O(n^3)$ steps. This assumes that all the matrix elements are constant. If, however the matrix ...
4
votes
2answers
44 views
Calculate the determinant when the sum of odd rows $=$ the sum of even rows
I have came across this interesting question in linear algebra and I couldn't know for sure the answer.
Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
4
votes
2answers
204 views
Elementary proof that if $A$ is $m \times n$ matrix map from $\mathbb{Z}^m$ then the map is surjective iff the gcd of det of minors is 1.
I am trying to find an elementary proof that if $\phi$ is a linear map from $\mathbb{Z}^n\rightarrow \mathbb{Z}^m$ represented by $A$, an $m \times n$ matrix the map is surjective iff the gcd*strong ...
4
votes
2answers
864 views
Coefficients of characteristic polynomial of a matrix
For a given matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed from the columns and rows with indices from $J$.
If the characteristic polynomial of A is ...
4
votes
3answers
68 views
Does assigning a different inner product to a vector space in $\mathbb{R^n}$ change the meaning of the determinant on that space?
We just started talking about inner product spaces and and how one can assign a different notion of length and angle on a vector space. Since the determinant in $\mathbb{R^n}$ captures the notion of ...
4
votes
1answer
387 views
Cross product of vectors as a determinant: valid matrix operation?
"The definition of the cross product can also be represented by the
determinant of a formal matrix."
—Wikipedia
This seems like a hack to me—something of much practical use but ...
4
votes
2answers
78 views
Determinants: A Special Condition
Under what conditions is
$$ \det(A_1 + \cdots + A_n) = \det(A_1)+\cdots+\det(A_n), $$
just curious.
4
votes
1answer
29 views
Determinant of the matrix $D_n(2,3,1)$
The matrix $D_n(2,3,1)$ is to be written in the form
$$\pmatrix{3 & 1 & 0 & 0 & ... & 0 \\ 2 & 3 & 1 & 0 & ... & 0 \\ 0 & 2 & 3 & 1 &... ...
