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PS1(2) If A, B = Mn(F) and a, BE F, prove that tr (aA + BB) = a tr A + 3 tr B.

PS1(2) If A, B = Mn(F) and a, BE F, prove that tr (aA + BB) = a tr A + 3 tr B.

Advanced Engineering Mathematics
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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Please follow the promot and use the facts provided only to prove the problem. Answer in latex please.

PS1(2) If A, B ≤ M₁ (F) and a, ß € F, prove that tr (aA + ßB) = a tr A + ß tr B.
Transcribed Image Text:PS1(2) If A, B ≤ M₁ (F) and a, ß € F, prove that tr (aA + ßB) = a tr A + ß tr B.
In this problem-set, F stands for C or R. In this problem-set, you are allowed to use the following facts: F1. If M € Mmxn(C), then the conjugate-transpose of M, denoted by M* = [m²¡], is the n-by-m matrix such that mij = mji. F2. If A € Mmxn (F) and B € Mnxp(F), then F3. If 2₁,..., Zn € C, then F4. If 21,..., Zn € C, then [AB]ij n n k=1 Σaikbkj. k=1 n 24-Ź² Zi = k=1 Σzi. n n II ²i = II Zi. k=1 k=1 F5. If A ≤ M₁ (F), then the trace of A, denoted by tr A = tr A, is defined by tr A = Σk_1 akk.
Transcribed Image Text:In this problem-set, F stands for C or R. In this problem-set, you are allowed to use the following facts: F1. If M € Mmxn(C), then the conjugate-transpose of M, denoted by M* = [m²¡], is the n-by-m matrix such that mij = mji. F2. If A € Mmxn (F) and B € Mnxp(F), then F3. If 2₁,..., Zn € C, then F4. If 21,..., Zn € C, then [AB]ij n n k=1 Σaikbkj. k=1 n 24-Ź² Zi = k=1 Σzi. n n II ²i = II Zi. k=1 k=1 F5. If A ≤ M₁ (F), then the trace of A, denoted by tr A = tr A, is defined by tr A = Σk_1 akk.
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