Question 6.5.4: What is the quadratic f = ax^2 + 2bxy + cy^2 for each of the...

Introduction to linear Algebra [EXP-672]

What is the quadratic $f = a{x}^{2} + 2bxy + c{y}^{2}$ for each of these matrices? Complete the square to write f as a sum of one or two squares ${d}_{1} (\quad )^{2} + {d}_{2} (\quad )^{2}$ .

$A =\begin{bmatrix} 1 & 2 \\ 2 & 9 \end{bmatrix}\quad$ and $\quad A=\begin{bmatrix} 1 & 3 \\ 3 & 9 \end{bmatrix}$ .

The Blue Check Mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Related Answered Questions

Test to see if R^TR is positive definite in each case: R= 1 2 0 3 and R= [ 1 1 1 2 2 1 ] and R= [ 1 1 2 1 2 1 ] .

Verified Answer:

{R}^{T}R =\begin{bmatrix} 1 & 2 \\ 2 &a...

Write f(x, y) = x^2 + 4xy + 3y^2 as a difference of squares and find a point (x, y) where f is negative. The minimum is not at (0,0) even though f has positive coefficients.

Verified Answer:

{x}^{2} + 4xy + 3{y}^{2} = (x + 2y)^{2} - {...

Suppose the 2 by 2 tests a > 0 and ac – b^2 > 0 are passed. Then c > b^2 a is also positive. (i) λ1 and λ2 have the same sign because their product λ1λ2 equals ____. (i) That sign is positive because λ1 + λ2 equals ____. Conclusion: The tests a > 0, ac – b^2 > 0 guarantee positive eigenvalues λ1,

Verified Answer:

Suppose a > 0 and

ac > {b}^{2}[/latex...

Suppose A^T = -A (real antisymmetric matrix). Explain these facts about A: (a) x^TAx = 0 for every real vector x. (b) The eigenvalues of A are pure imaginary. (c) The determinant of A is positive or zero (not negative). For (a), multiply out an example of x^TAx and watch terms cancel. Or reverse x^T

Verified Answer:

(a)

{x}^{T}(Ax) = (Ax)^{T}x = {x}^{T}{A}^{T...

If λmax is the largest eigenvalue of a symmetric matrix A, no diagonal entry can be larger than λmax. What is the first entry a11 of A = QΛQ^T? Show why a11 ≤ λmax.

Verified Answer:

{a}_{11}

is

\left[ \begin{ma...

This A is nearly symmetric. But its eigenvectors are far from orthogonal: A = [ 1 10^-15 0 1+10^-15 ] has eigenvectors [ I 0 ] [?] What is the angle between the eigenvectors?

Verified Answer:

Eigenvectors (1, 0) and (1, 1) give a

{45}^...

Find all 2 by 2 matrices that are orthogonal and also symmetric. Which two numbers can be eigenvalues?

Verified Answer:

Orthogonal and symmetric requires

\left| \l...

What number b in [ 2 b 1 0 ] makes A = QΛQ^T possible? What number makes A = SΛS^-1 impossible? What number makes A^-1 impossible?

Verified Answer:

Symmetry gives

Q\wedge {Q}^{T}

if b...

(Recommended) Which of these classes of matrices do A and B belong to: Invertible, orthogonal, projection, permutation, diagonalizable, Markov? A= [ 0 0 1 0 1 0 1 0 0 ] B= 1/3 [ 0 0 1 0 1 0 1 0 0 ] . Which of these factorizations are possible for A and B: LU, QR, SΛS^-1, QΛQ^T?

Verified Answer:

A is invertible, orthogonal, permutation, diagonal...

(A paradox for instructors) If AA^T = A^T A then A and A^T share the same eigenvectors (true). A and A^T always share the same eigenvalues. Find the flaw in this conclusion: They must have the same S and Λ. Therefore A equals A^T.

Verified Answer:

A and

{A}^{T}

have the same

...