# Proving a linear algebra equation

• IlyaMath
In summary, the conversation discusses the difficulty in proving the equivalence of two multivariate formulas and the use of MATLAB to numerically verify their equivalence. The formulas, A and B, involve various matrices and variables π, y, q, β, Σ, Σπ, Σy, Σβ, Σε, X, P, and τ, with assumptions and notations provided. The conversation also mentions that the problem comes from Bayesian statistics and the importance of the values of N and F in the proof.
IlyaMath
I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent.

I would appreciate any help on this.

Prove A = B

A = (Σπ^-1 + Σy^-1)^-1 * (Σπ^-1*π + Σy^-1*y)

y = π+ X*β

Σπ =τ*Σ

Σy = X' * Σβ * X + ΣεB = (Σπ^-1 + P'*Σβ^-1*P)^-1 * (Σπ^-1*π + P'*Σβ^-1*q)

q = P*y

P = (X'*Σ^-1*X)^-1*X'*Σ^-1Assumptions

i) Σε is infinitesimally small. (If Σε is exactly zero, then Σy may not be invertible).

ii) N > F (If N = F, then the proof is trivial. If N < F then is probably not invertible and P is not defined.)Notation

A, B: Nx1

π: Nx1

y: Nx1

q: Fx1

β: Fx1

Σ: NxN

Σπ: NxN

Σy: NxN

Σβ: FxF

Σε: NxN

X: NxF

P: FxN

τ: 1x1

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"Sigma", $\Sigma$, here, is a matrix, not a summation symbol?

Yes, HallsofIvy, you are correct...

This problem comes from Bayesian statistics, where N is number of observations and F is number of factors. All of the Sigma (Σ) are variance-covariance matrices. The Greek characters after each Σ are meant to be subscripts.

## 1. How do I prove a linear algebra equation?

To prove a linear algebra equation, you can use a variety of methods such as direct proof, contrapositive proof, or proof by contradiction. It is important to follow the rules of algebra and logic to ensure a valid proof.

## 2. What is the difference between a vector and a scalar?

A vector is a quantity that has both magnitude and direction, while a scalar is a quantity that only has magnitude. In linear algebra, vectors are represented by arrows and can be added, subtracted, and multiplied by a scalar.

## 3. Can I use matrices to prove linear algebra equations?

Yes, matrices can be used to represent linear algebra equations and can be manipulated using various operations such as addition, subtraction, and multiplication. Matrices can also be used to solve systems of linear equations.

## 4. How do I know if an equation is linear or not?

A linear equation is an equation that has a degree of one and follows the form y = mx + b, where m is the slope and b is the y-intercept. You can also graph the equation and see if it forms a straight line. If it does, then it is a linear equation.

## 5. Can I use linear algebra to solve real-world problems?

Yes, linear algebra has many practical applications in fields such as engineering, physics, economics, and computer science. It can be used to solve problems involving optimization, data analysis, and modeling real-world scenarios.

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