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Chapter 6: Problem 13

Let \(A\) be a symmetric positive definite \(n \times n\) matrix and let \(S\) be a nonsingular \(n \times n\) matrix. Show that \(S^{T} A S\) is positive definite

Short Answer

Expert verified
In summary, we showed that the quadratic form \(Q_B(x) = x^{T}(S^T A S)x = y^{T} A y\), where \(y = Sx\), is positive for any non-zero vector x. Since A is positive definite and S is nonsingular, we concluded that B = \(S^T A S\) is also positive definite.

Step by step solution

01

Define the quadratic form for B

Let x be a non-zero vector in 鈩漗n. Define the quadratic form for the matrix B as \(Q_B(x) = x^{T} B x = x^{T} (S^T A S) x\).
02

Substitute B

Now, substitute the expression for B. We have: \(Q_B(x) = x^{T} (S^T A S) x\).
03

Rearrange terms

Next, we will rearrange the terms in the quadratic form to isolate A: \(Q_B(x) = (x^{T} S^T) A (Sx)\).
04

Introduce new variable

Introduce a new variable, y, such that \(y = Sx\). Since S is a nonsingular matrix, this transformation is a one-to-one mapping and y is a non-zero vector if x is non-zero. Now, the quadratic form becomes: \(Q_B(x) = y^{T} A y\).
05

Positive definiteness of A

Given that A is positive definite, its quadratic form with respect to y is positive for any non-zero vector y: \(y^{T} A y > 0\).
06

Conclude positive definiteness of B

Since we have shown that the quadratic form of B is positive for any non-zero vector x, this implies that B is positive definite.

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Most popular questions from this chapter

Show that \(A\) and \(A^{T}\) have the same nonzero singular values. How are their singular value decompositions related?

Use the definition of the matrix exponential to compute \(e^{A}\) for each of the following matrices: (a) \(A=\left(\begin{array}{rr}1 & 1 \\ -1 & -1\end{array}\right)\) (b) \(A=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)\) (c) \(A=\left(\begin{array}{rrr}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)\)

Find the general solution of each of the following systems: (a) \(y_{1}^{\prime \prime}=-2 y_{2}\) (b) \(y_{1}^{\prime \prime}=2 y_{1}+y_{2}^{\prime}\) \(y_{2}^{\prime \prime}=y_{1}+3 y_{2} \quad y_{2}^{\prime \prime}=2 y_{2}+y_{1}^{\prime}\)

Let \\[ A=\left(\begin{array}{rrrr} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array}\right) \\] (a) Compute the \(L U\) factorization of \(A\) (b) Explain why \(A\) must be positive definite.

Let \(A\) be a Hermitian matrix and let \(B=i A\). Show that \(B\) is skew Hermitian.
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