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Chapter 8: Q19E (page 413)

All positive definite matrices are invertible.

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

It is known that a matrix is positive definite if its Eigenvalues are positive.

Also, there are no entries in the diagonal zero. So, no row of the given matrix can be made zero which means all the row of the matrix are linearly independent.

Moreover, a matrix A is not invertible if 0 is an Eigen value of A.

Now,A is positive definite implies all Eigen values of A are positive, in particular they are non-zero. Hence, must be invertible.

02

Final Answer

The given statement is TRUE.

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Most popular questions from this chapter

If A is an n脳nmatrix, what is the product of its singular values 蟽1,...,蟽n? State the product in terms of the determinant of A. For a 2脳2matrix A, explain this result in terms of the image of the unit circle.

LetLfrom R3to R3be the reflection about the line spanned by

v鈫�=[102]

a. Find an orthonormal eigenbasis for L .

b. Find the matrix B of L with respect to eigenbasis.

c. Find the matrix A of L with respect to the standard basis ofR3 .

Show that any matrix of rank rcan be written as the sum of r matrices of rank 1.

If A is a symmetric n x n matrix, what is the relationship between the eigenvalues of A and the singular values of A?

Cholesky factorization for 2脳2 matrices. Show that any positive definite 2脳2 matrix A can be written uniquely A=LLT as where L is a lower triangular2脳2 matrix with positive entries on the diagonal. Hint: Solve the equation

[abbc]=[x0yz][xy0z]
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