TMA4267 Linear Statistical Models - useful rules in matrix algebra by Mette Langaas

TMA4267 Linear Statistical Models - useful rules in matrix algebra by Mette Langaas https://padlet.com/mette_langaas/t0pfmgt4jqad Write down your favourite rule - to help yourself and all class mates get an overview of useful rules to be used in the course. en-us 2017-01-21 15:17:00 UTC 2017-05-15 16:51:24 UTC hello@padlet.com https://padlet-assets.s3.amazonaws.com/icons/Thunder.png The trace tr() of a quadratic matrix is the sum of the diagonal elements. mette_langaas https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148517811 Let A, B and C be conformable matrices, then tr(AB)=tr(BA) and tr(ABC)=tr(BCA)=tr(CAB).
Proof: look at the diagonal elements and see at the sum of these are equal for the formulas.

]]> 2017-01-21 15:22:56 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148517811 Determinants nilsbar https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148520004 $$ |A^T| = |A| $$
$$|AB| = |A| \cdot |B| $$
$$ |cA| = c^n|A| $$
$$\lvert A^{-1}\rvert=\frac{1}{\lvert A \rvert}$$]]> 2017-01-21 16:09:47 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148520004 Orthogonal matrices nilsbar https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148520390 A^T=A^{-1}

and

A^TA=AA^T=I]]> 2017-01-21 16:17:46 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148520390 A is a square p-dimensional real and diagonalizable matrix. mette_langaas https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148537517 We write $$A=P\Lambda P^T$$ where P is an orthogonal matrix with the eigenvectors $$e_i$$
of A as column vectors and $\Lambda$ is a diagonal matrix with the eigenvalues $\lambda_i$ of A on the diagonal.
$$\mbox{tr}(A)=\sum_{i=1}^p \lambda_i$$
$$\mbox{det}(A)=\prod_{i=1}^p \lambda_i$$

]]> 2017-01-22 00:20:05 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/148537517 Idempotent matrices jakobpeder_pettersen https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/149105811 Definition: A square matrixA

is idempotent if:

A^2=AA=A

Assume

A

is idempotent. Then the following hold:

$$\text{If $\lambda$ is an eigenvalue of $A$, then $\lambda=0$ or $\lambda=1$ }$$
$$\mathrm{rank}(A)=\mathrm{tr}(A)$$
$$\text{For any vector $\mathbf{x}\in\mathbb{R}^n$,}$$ $$\text{$A\mathbf{x}$ is the projection of $\mathbf{x}$ onto the column space of $A$}$$

]]> 2017-01-24 17:52:21 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/149105811 Transpose scottbun https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/152703492
(AB)^T = B^TA^T]]> 2017-02-09 08:59:20 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/152703492 Trace indentities user8715 https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/156979209
$$ tr(cA) = c tr(A) $$
$$ tr(A + B) = tr(A) + tr(B) $$
$$ tr(A) = tr(A^T) $$
$$ tr(AB) = tr(BA) = \sum_{i,j} A_{i,j}B_{i,j}$$

Trace formula:
$$E[X^TAX]=tr(AΣ)+\mu^TA\mu$$

]]> 2017-03-01 11:31:53 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/156979209 Transpose rasmusmunter https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/171660856 (AB)^T = B^TA^T]]> 2017-05-14 09:25:00 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/171660856 Hvis man 'highlighter' teksten så kommer det opp en meny med Pi. Trykk på den og dropp $$ tegnene så funker det! rasmusmunter https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/171660897 Takk Rasmus - da er det kommet på plass siden 12.05 - bra - tusen takk for hjelpen! Mette]]> 2017-05-14 09:26:00 UTC https://padlet.com/mette_langaas/t0pfmgt4jqad/wish/171660897