Laplace expansion is a method for expanding determinants in terms of minors, determinants of some related smaller matrices.
Laplace expansion is a method for expanding determinants in terms of minors, determinants of some related smaller matrices.
Questions tagged [laplace-expansion]
63 questions
11
votes
2 answers
Proof of Leibniz formula from Laplace expansion
I'm trying to prove Leibniz formula for the determinant using Laplace expansion. Here's my attempt:
For a $1 \times 1$ matrix $A = \begin{pmatrix}a_{11}\end{pmatrix}$, define $\det A = a_{11}$. For any $n \times n$ matrix $A$, define $\det A$…
David
- 1,252
9
votes
5 answers
How to compute the determinant of a Kac-Murdock-Szegő (KMS) Toeplitz matrix?
Given a positive integer $n$, express$$
f_n(x) = \left|\begin{array}{c c c c c}
1 & x & \cdots & x^{n - 1} & x^n\\
x & 1 & x & \cdots & x^{n - 1} \\
\vdots & x & \ddots & \ddots & \vdots\\
x^{n - 1} & \vdots & \ddots & 1 & x\\
x^n & x^{n - 1} &…
Ѕᴀᴀᴅ
- 35,369
6
votes
1 answer
Is this matrix positive-semidefinite in general?
for the matrix written below I was wondering if one can show that it is positive-semidefinite for $n>3$ and $0< \alpha<1$. (Or not. For $n=2, 3$ it works by showing that all principal minors are non-negative.)
$$
C_{n,n} =
\begin{pmatrix}
1 &…
JoSt
- 63
6
votes
1 answer
Proving:$\operatorname{Proj}_{U^\perp}(x)=-\frac1{\det(A^TA)} X(u_1,\ldots, u_{n-2}, X(u_1,\ldots, u_{n-2}, x))$
The problem I'm trying to solve is as follows, which was posed to me by my professor as an exercise:
Let $x, u_i \in \Bbb R^n$, $ A = (u_1, u_2, \ldots, u_{n-2})$ and $\{u_1, u_2, \ldots, u_{n-2}\}$ is linearly independent. Let $U = \text{Col}(A)$.…
learning_linalg
- 101
6
votes
2 answers
Help me with the result of this determinant..
$$
D =
\begin{vmatrix}
1 & 1 & 1 & \dots & 1 & 1 \\
2 & 1 & 1 & \dots & 1 & 0 \\
3 & 1 & 1 & \dots & 0 & 0 \\
\vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\
n-1 & 1 & 0 & \dots & 0 & 0 \\
n & 0 & 0 & \dots & 0 & 0…
A6EE
- 313
4
votes
2 answers
Determinant of 4x4 Matrix by Expansion Method
Find det(B) = \begin{bmatrix} 2 & 5 & -3 & -2 \\ -2 & -3 & 2 & -5 \\ 1 & 3 & -2 & 0 \\ -1 & -6 & 4 & 0 \\ \end{bmatrix}
I chose the 4th column because it has the most 0s. Using basketweave, I solved for the determinants of the minor 3x3 matrices of…
k_rollo
- 143
4
votes
2 answers
Proof of the Laplace Expansion?
I just learned about Laplace Expansion for determinant calculation in high-school, they taught me how to calculate minors cofactors and everything but they did not include the proof in the book. I have tried watching YouTube videos and there is a…
Mick_Mick
- 61
- 1
- 5
4
votes
1 answer
Calculating general $n\times n$ determinant
I'm given determinant $\begin{vmatrix}
1 & 2 &3 & \cdots & n -1 & n \\
2 & 3 &4 & \cdots & n & 1 \\
3 & 4 &5 & \cdots & 1 & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
n & 1 & 2 & \cdots & n-2 & n-1
\end{vmatrix}$ and I have to…
3
votes
2 answers
Is there an easy way for a person to compute the determinant of an arbitrary matrix?
I'm having a tough time proving statements that involve the determinant of an arbitrary matrix and was wondering if there is just an easier way to compute it or some simpler equivalent definition for it. Just to give an example:
Let $K$ be a field,…
Alp
- 748
3
votes
0 answers
prove the determinant identities
Let $M$ be an $n \times n$ matrix, $N = \{1, \dots , n \}$ and $V \subseteq N$. Let $M^{(V)}$ denote the submatrix $(m_{i,j})$ of $M$ with $i,j \in V$. Prove that
$$\det(Y-\iota I_n)^{(U)}=\sum_{W\subseteq U} (-\iota)^{card(W)} \det(Y^{(U-W)})$$…
Manish Saini
- 590
- 3
- 12
3
votes
0 answers
what's wrong with this matrix (to find the determinant using Laplace expansion)?
I have to compute the determinant of this 4x4 matrix:
\begin{bmatrix}2&1&3&0\\-1&0&1&2\\2&0&-1&-1\\-3&1&0&1\end{bmatrix}
this is what I did:
I swapped first column with second column (and according to linear algebra theory, the determinant becomes…
3
votes
2 answers
Asymptotic expansion for the following integral
I was trying to find the asymptotic expansion for
$$\int_0^1 \sqrt{t(1-t)}(t+a)^{-x} \; \mathrm{d}t,$$ for $a>0$ as $x \rightarrow \infty$. I have already tried re-writing $$(t+a)^{-x}=\exp(- x\log(t+a)).$$ However, by using Laplace's method for…
Petals
- 53
3
votes
1 answer
Laplace expansion
This statement is from the book of Winitzki Linear Algebra via Exterior Products. (Section 3.4, page 123) Let $V$ be finite dimensional vector space, $\dim(V)=N$. The determinant of the matrix $v_{ij}$ is a linear function of each of the vectors…
vesszabo
- 3,563
3
votes
1 answer
What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?
It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation to minors of $A$?
I have tried some cases…
Troy Woo
- 3,647
2
votes
1 answer
Proof of Laplace expansion using minors
I've come across with the following proof of the Laplace expansion:
Let
$\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$
and
$\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar M_j^i$
We'll prove by induction that $\tilde{\Delta}=\Delta$.
For a…
user175788
- 55