Another cool application of summation notation with matrices is to prove things about the trace of a matrix. The trace only applies to square matrices (equal number of rows and columns) and is the sum of all the entries on the diagonal-that is, the sum of all entries with the same column and row number. In summation notation, the trace of an n x n matrix as: Tr (A) = a1,1 + a2,2+...+ ann =Σaii This time we are using the index i for both the row position and the column position, so its the position of the index that denotes row and column. The formula for the product used two different letters for the indices because they were not always equal, but for trace the row and column number will always be equal, so we only need one letter. The next exercise covers some basic properties of traces: Exercise 11.7.1. (a) Prove that if A and B are square matrices of the same size, then Tr (A + B) = Tr (A) + Tr (B). (b) Prove that if A is a square matrix with real entries and k is a real number, then Tr (kA) = kTr (A).
Another cool application of summation notation with matrices is to prove things about the trace of a matrix. The trace only applies to square matrices (equal number of rows and columns) and is the sum of all the entries on the diagonal-that is, the sum of all entries with the same column and row number. In summation notation, the trace of an n x n matrix as: Tr (A) = a1,1 + a2,2+...+ ann =Σaii This time we are using the index i for both the row position and the column position, so its the position of the index that denotes row and column. The formula for the product used two different letters for the indices because they were not always equal, but for trace the row and column number will always be equal, so we only need one letter. The next exercise covers some basic properties of traces: Exercise 11.7.1. (a) Prove that if A and B are square matrices of the same size, then Tr (A + B) = Tr (A) + Tr (B). (b) Prove that if A is a square matrix with real entries and k is a real number, then Tr (kA) = kTr (A).
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 do part A and B and please show step by step and explain
Transcribed Image Text:Another cool application of summation notation with matrices is to prove
things about the trace of a matrix. The trace only applies to square matrices
(equal number of rows and columns) and is the sum of all the entries on
the diagonal-that is, the sum of all entries with the same column and row
number. In summation notation, the trace of an n x n matrix as:
Tr (A) = a1,1 + a2,2+...+ ann =Σaii
This time we are using the index i for both the row position and the column
position, so its the position of the index that denotes row and column. The
formula for the product used two different letters for the indices because
they were not always equal, but for trace the row and column number will
always be equal, so we only need one letter.
The next exercise covers some basic properties of traces:
Exercise 11.7.1.
(a) Prove that if A and B are square matrices of the same size, then
Tr (A + B) = Tr (A) + Tr (B).
(b) Prove that if A is a square matrix with real entries and k is a real
number, then Tr (kA) = kTr (A).
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