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Chapter 7: Q25E (page 395)

Show that if \(B\) is \(m \times n\), then \({B^T}B\) is positive semidefinite; and if \(B\) is \(n \times n\) and invertible, then \({B^T}B\) is positive definite.

Short Answer

Expert verified

It is proved that if \(B\) is \(m \times n\), then \({B^T}B\) is positive semidefinite; and if \(B\) is \(n \times n\) and invertible, then \({B^T}B\) is positive definite.

Step by step solution

01

Step 1: Find the characteristic polynomial

We need to prove that\({B^T}B\)is a symmetric matrix and the quadratic form\({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} \ge 0\)for all\({\bf{x}}\)that proves\({B^T}P\)positive semi-definite.

Now consider,

\(\begin{aligned}{}{\left( {{B^T}B} \right)^T} &= {B^T}{\left( {{B^T}} \right)^T}\\ &= {B^T}B\end{aligned}\)

Thus, the matrix \({B^T}B\) is a symmetric matrix.

02

Show that \({B^T}B\) is positive semi definite

Consider the quadratic form \({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}}\).

\(\begin{aligned}{}{{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} &= \left( {{{\bf{x}}^T}{B^T}} \right)B{\bf{x}}\\ &= {\left( {B{\bf{x}}} \right)^T}B{\bf{x}}\\ &= {\left\| {B{\bf{x}}} \right\|^2}\end{aligned}\)

Thus, \({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} \ge 0\).

Therefore, the matrix \({B^T}B\) is positive semi-definite.

03

Show that \({B^T}B\) is positive definite

Assume \({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} = 0\)then we have,

\(\begin{aligned}{}{{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} &= 0\\\left( {{{\bf{x}}^T}{B^T}} \right)B{\bf{x}} &= 0\\{\left( {B{\bf{x}}} \right)^T}B{\bf{x}} &= 0\\{\left\| {B{\bf{x}}} \right\|^2} &= 0\end{aligned}\)

It is given that the matrix\(B\)is invertible, hence the equation\(B{\bf{x}} = 0\)has a trivial solution only. Therefore, we have,\({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} = 0\)then\({\bf{x}} = 0\)and\({\bf{x}} \ne 0\),\({{\bf{x}}^T}\left( {{B^T}B} \right){\bf{x}} > 0\).

Thus,\({B^T}B\)is a positive definite.

It is proved that if \(B\) is \(m \times n\), then \({B^T}B\)is positive semidefinite; and if\(B\) is \(n \times n\) and invertible, then \({B^T}B\)is positive definite.

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Most popular questions from this chapter

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

16. \(\left( {\begin{aligned}{{}}{\,6}&{ - 2}\\{ - 2}&{\,\,\,9}\end{aligned}} \right)\)

In Exercises 17鈥�24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

17. Show that if \(A\) is square, then \(\left| {{\bf{det}}A} \right|\) is the product of the singular values of \(A\).

Question: Mark Each statement True or False. Justify each answer. In each part, A represents an \(n \times n\) matrix.

  1. If A is orthogonally diagonizable, then A is symmetric.
  2. If A is an orthogonal matrix, then A is symmetric.
  3. If A is an orthogonal matrix, then \(\left\| {A{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
  4. The principal axes of a quadratic from \({{\bf{x}}^T}A{\bf{x}}\) can be the columns of any matrix P that diagonalizes A.
  5. If P is an \(n \times n\) matrix with orthogonal columns, then \({P^T} = {P^{ - {\bf{1}}}}\).
  6. If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
  7. If \({{\bf{x}}^T}A{\bf{x}} > {\bf{0}}\) for some x, then the quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is positive definite.
  8. By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
  9. The largest value of a quadratic form \({{\bf{x}}^T}A{\bf{x}}\), for \(\left\| {\bf{x}} \right\| = {\bf{1}}\) is the largest entery on the diagonal A.
  10. The maximum value of a positive definite quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is the greatest eigenvalue of A.
  11. A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable \({\bf{x}} = P{\bf{u}}\), for some orthogonal matrix P.
  12. An indefinite quadratic form is one whose eigenvalues are not definite.
  13. If P is an \(n \times n\) orthogonal matrix, then the change of variable \({\bf{x}} = P{\bf{u}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form whose matrix is \({P^{ - {\bf{1}}}}AP\).
  14. If U is \(m \times n\) with orthogonal columns, then \(U{U^T}{\bf{x}}\) is the orthogonal projection of x onto ColU.
  15. If B is \(m \times n\) and x is a unit vector in \({\mathbb{R}^n}\), then \(\left\| {B{\bf{x}}} \right\| \le {\sigma _{\bf{1}}}\), where \({\sigma _{\bf{1}}}\) is the first singular value of B.
  16. A singular value decomposition of an \(m \times n\) matrix B can be written as \(B = P\Sigma Q\), where P is an \(m \times n\) orthogonal matrix and \(\Sigma \) is an \(m \times n\) diagonal matrix.
  17. If A is \(n \times n\), then A and \({A^T}A\) have the same singular values.

Question: 12. Exercises 12鈥�14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Verify the properties of\({A^ + }\):

a. For each\({\rm{y}}\)in\({\mathbb{R}^m}\),\(A{A^ + }{\rm{y}}\)is the orthogonal projection of\({\rm{y}}\)onto\({\rm{Col}}\,A\).

b. For each\({\rm{x}}\)in\({\mathbb{R}^n}\),\({A^ + }A{\rm{x}}\)is the orthogonal projection of\({\rm{x}}\)onto\({\rm{Row}}\,A\).

c. \(A{A^ + }A = A\)and \({A^ + }A{A^ + } = {A^ + }\).

Suppose A is invertible and orthogonally diagonalizable. Explain why \({A^{ - {\bf{1}}}}\) is also orthogonally diagonalizable.
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