Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A) rank(B) = b, and a > b. = a, a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and to the null space of B. b) Based on a), which of the following statements is true? a. rank (A+B) = a+b b. rank(A+B) > a c. rank (A+B) < a c) What is the rank of the matrix A²
Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A) rank(B) = b, and a > b. = a, a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and to the null space of B. b) Based on a), which of the following statements is true? a. rank (A+B) = a+b b. rank(A+B) > a c. rank (A+B) < a c) What is the rank of the matrix A²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A)
rank(B) = b, and a > b.
= a,
a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and
to the null space of B.
b) Based on a), which of the following statements is true?
a. rank (A+B) = a+b
b. rank(A+B) > a
c. rank (A+B) < a
c) What is the rank of the matrix A²
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