# Questions tagged [determinant]

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### How to write or show statements for a matrix of order 2 or more [migrated]

Let $A$ be an $n \times n$ orthogonal matrix where $n$ is even, $|A|=-1$. Show that $|I-A|=0$. I can prove this statement for a matrix of $n=2$, I can slog and also provide a proof for $n=4$, but how ...
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### How to prove the determinant of covariance matrix is zero, when n≤p?

If there are n observations on p dimensions, then the covariance matrix will be: But when n≤p, its determinant will be zero. I know it is because it becomes as a singular matrix, but I do not know ...
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### What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?

What is the meaning of the quantity: $$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$ Is there, for example, a geometric explanation? Is there a term for it in statistics?
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### GDA producing negative covariance determinant

I'm running Gaussian Discriminant Analysis across a large set of examples (~80k) in $\mathbb{R}^{8}$. I know that the covariance matrix $\Sigma$ is, by definition, positive semi-definite, which means ...
36 views

### Linear dependency among columns and rows

Singular matrix is defined as square matrix with the determinant of zero. The determinant of zero occurs when matrix columns are linearly dependent (i.e. one of the columns can be defined as a linear ...
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### Singular Matrix and Linear Dependency

Singular matrix is defined as a square matrix with determinant of zero. I am aware that linear dependency among columns or rows leads to determinant being equal to zero (e.g. one column is a linear ...
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### How to find optimum matrix set based on determinant values using python

I am new at programming, so I want to find the optimum set of row values based on maximum determinant logic. 1) Set the 1st Column 'Serial_no' as index. 2) Take first 'N' row values as user input ...
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### Why does the determinant of the Hessian grow with n?

Context: I'm trying to understand BIC on a deeper level. I'm using BIC for model/structure selection for Bayesian networks. I'm confused because BIC is an approximation to the likelihood of a model, ...
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### How to model for independent determinants in several groups based on follow-up time

I want to answer the research question which determinants are associated with long-term survival after a myocardial infarction (MI) in a prospective patient cohort study. More precisely: I want to ...
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### Do the Determinants of Covariance and Correlation Matrices and/or Their Inverses Have Useful Interpretations?

While learning to calculate covariance and correlation matrices and their inverses in VB and T-SQL a few years ago, I learned that the various entries have interesting properties that can make them ...
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### Multicollinearity (or not) in exploratory factor analysis

I’m performing an exploratory factor analysis with 28 items, n = 300. I’m confused whether I have a multicollinearity problem or not, and if so whether/how I go about choosing items to remove from the ...
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### Algebraic derivation of canonical correlations

In a paper from 1936, Harold Hotelling (access on JSTOR) defined the concepts of canonical correlations and canonical variates for two sets of variates. In pages 327 and 328, he precisely derives ...
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### Maximum determinant with $l_2$ norm penalties

Let the random sample $Y_1,\ldots,Y_n\sim N_p(0,\Sigma)$, then the likelihood is given by, \begin{align*} L(\Sigma)=\frac{1}{(2\pi)^{np/2}|\Sigma|^{n/2}} e^{-\frac{1}{2}\sum_i Y'_i\Sigma^{-1}Y_i} \end{...
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Let $\Lambda \sim \mathcal W(\nu, \Psi)$, i.e., following a $n \times n$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. The expectation of the log determinant of $\... 1answer 139 views ### Determinant of a block matrix with sparse elements I have a positive definite symmetric matrix that looks like $$\pmatrix{A & 0 & 0 & E \\ 0 & B & 0 & F \\ 0 & 0 & C & G \\ E^\prime & F^\prime & G^\prime &... 1answer 209 views ### How to get the determinant of a covariance matrix from its diagonal elements I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ... 0answers 115 views ### Stability of VAR(p): Prove \det(I_{Kp}-\tilde{A}z)=\det(I_k-A_1z-\ldots-A_pz^p) Given we have a VAR(p) process written in the companion form$$\tilde{y}=\tilde{v}+\tilde{A}\tilde{y}_{t-1}+\tilde{u}_t$$where$$\tilde{A}=\left(\begin{array}{ccccc} A_1& A_2 & \ldots &... 1answer 826 views ### How is the determinant of$(X'X)$related to variance? I'm working on a problem (and actually have the answer) but I don't know why this is the answer, can someone explain this equality?. It has to do with the the determinant of the partitioned matrix$(X'...
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I'm working on my homework and I keep seeing something like $$(x'x-x'W(W'W)^{-1}W'x)$$ I know that $W(W'W)^{-1}W'$ is the projection matrix, but what is so special about subtracting those two inner ...
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### Column1 = Temp_F and Column2 = Temp_C — are these linearly dependent?

If $X$ is a matrix of size $m$ x $2$, where the first column is a range of Celsius values and the second column is their corresponding Fahrenheit values, would the columns be considered linearly ...
Suppose there is a structural VAR model, such as: $A y_{t} = B y_{t-1} + \varepsilon_{t}$, where $\varepsilon_{t} \sim N(0, I_{n})$; then the matrix representing the contemporaneous impulse-response ...