Show that any positive definite n ×n matrix A can be written as A=B B^T, where B is an n ×n matr

step 1 For this exercise, we have that D sub 1 can equal 3 .002. And we have S equal to 1 over root 5,

Show that any positive definite n ×n matrix A can be written...

Show that any positive definite $n \times n$ matrix $A$ can be written as $A=B B^{T},$ where $B$ is an $n \times n$ matrix with orthogonal columns. Hint: There exists an orthogonal matrix $S$ such that $S^{-1} A S=S^{T} A S=D$ is a diagonal matrix with positive diagonal entries. Then $A=S D S^{T}$ Now write $D$ as the square of a diagonal matrix.

Key Concepts

Positive Definite Matrix

A positive definite matrix is a symmetric matrix for which all eigenvalues are positive. This property guarantees that the quadratic form x?Ax is positive for all nonzero vectors x, which is crucial for many applications and allows various decompositions such as the one in this problem.

Spectral Theorem / Diagonalization

The spectral theorem states that any real symmetric matrix can be diagonalized by an orthogonal matrix. This means that there exists an orthogonal matrix S such that S?AS is a diagonal matrix D. This concept is key in transforming the original matrix into a simpler form to facilitate factorization.

Orthogonal Matrix

An orthogonal matrix has the property that its transpose is its inverse, meaning S?S = I. This characteristic ensures that the transformation preserves lengths and angles, making the diagonalization process stable and reliable. It is also important in constructing matrices with orthogonal columns as required in the factorization A = B B?.

Matrix Square Root and Factorization

Given a diagonal matrix D with positive entries, one can easily define its square root by taking the square root of each diagonal entry. This square root matrix, denoted D^(1/2), satisfies D = D^(1/2) D^(1/2). By substituting this into the diagonalized form of A, one can express A as the product of a matrix with orthogonal columns and its transpose. This process underlies several matrix decomposition techniques used in various fields.

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