Strang §2.7 · Transposes
The transpose swaps rows and columns.
Row 1 of A becomes column 1 of Aᵀ. An m by n matrix turns into an n by m matrix. A matrix that equals its own transpose is called symmetric.
123456ᵀ = 135246
Think: turning a spreadsheet
A sales sheet has products as rows and months as columns. Your manager wants months as rows instead. You haven't changed any number, only which way you read them. That is the transpose.
Symmetric: the distance table
Look at a road atlas table of distances between cities. Paris to Lyon is the same as Lyon to Paris. Flip the table across its diagonal and nothing changes. That table is symmetric: S = Sᵀ.
The rules
- (AB)ᵀ = BᵀAᵀ. The order flips, just like the inverse.
- (A⁻¹)ᵀ = (Aᵀ)⁻¹. Transpose and inverse can be done in either order.
- Dot products: x · y = xᵀy, a row times a column.
- AᵀA is always symmetric. Its (i, j) entry is column i of A dotted with column j. Swapping i and j gives the same dot product.
Worked example
A = 123456 is 3 by 2, so AᵀA is 2 by 2.
(1,1): 1 + 9 + 25 = 35 (1,2): 2 + 12 + 30 = 44 (2,2): 4 + 16 + 36 = 56
AᵀA = 35444456
Where you meet this
AᵀA is the heart of least squares (chapter 4): fitting a line to noisy measurements, or calibrating an accelerometer from many readings. For a rotation matrix Q, QᵀQ = I, so the transpose is the inverse. Turning back is free.