Prove that ⟨A, B⟩=tr(A^T B) defines an inner product on the vector space ℳn ×n of real n ×n matr

Prove that ⟨A, B⟩=tr(A^T B) defines an inner product on the...

Key Concepts

Inner Product

An inner product is a function on a vector space that takes in two vectors and returns a scalar, satisfying linearity, symmetry, and positive definiteness. It provides a way to define geometric concepts such as lengths and angles within that space, ultimately giving the space its structure as an inner product space.

Matrix Transpose

The transpose of a matrix is obtained by swapping its rows and columns. It is fundamental in many linear algebra contexts because it helps establish properties like symmetry and is essential in forming expressions, such as A^T B, that are used in definitions of inner products on matrix spaces.

Trace of a Matrix

The trace of a square matrix is defined as the sum of its diagonal elements. It is a linear operator and possesses invariance properties under cyclic permutations, making it a valuable tool in many areas of linear algebra, including the formulation of inner products in spaces of matrices.

Frobenius Inner Product

The Frobenius inner product is a specific inner product defined on the space of matrices using the trace function. It is given by the formula ?A, B? = tr(A^T B) and encapsulates the properties of symmetry, linearity, and positive definiteness, thereby making it a standard way to measure angles and lengths in matrix spaces.

Transcript

00:05 We're told that v is a vector space of m by n matrices over the field of real numbers.

00:19 We're asked to show that the function defined by inputs a and b is the trace of b transpose a defines an inner product in v.

00:43 Maybe the best way to do this is to treat this trace.

00:50 This is the sum from i equals 1 to m, the sum from j equals 1 to n of a.

01:01 A .i .j.

01:07 B .i .j.

01:18 To show that this defines an inner product, we'll see if it satisfies the inner product axioms.

01:28 So first we'll prove positive definiteness.

01:35 Actually, first i'll prove property i1.

01:37 The linear property, well, let a and b be real numbers, and we'll let a1, a2, and b the m by n matrices over the real numbers.

02:03 Then the inner product of a times a1 plus b times a2 with b is equal to, well, this is the trade.

02:19 Of b transpose times a 1 plus b times a 2 and this is equal to the trace of distributing b transpose times a 1 plus b transpose times b times a 2 which of course is the same as the trace of a times b transpose a 1 plus b times b transpose a 1 plus b times b transpose a2, and we know that the trace of a sum of matrices is the same as the sum of the traces of matrices.

03:03 This is the same as the trace of a times b transpose a1, plus the trace of b times b transpose a2, which we know that the trace of a scalar times a matrix is the same as the scalar times the trace of the matrix.

03:25 This is a times the trace of b transpose a1 plus b times the trace of b transpose a2.

03:34 And by definition, this is a times the inner product of a1 with b, plus b times the interproduct of a2 with b...

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