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Transpose &
Symmetric Matrices

Master the art of flipping rows and columns, discover when a matrix equals its own transpose, and unlock powerful properties used across linear algebra.

Linear Algebra

What Is a Transpose?

The transpose of a matrix \( A \), written \( A^T \), is formed by swapping rows and columns. The element in row \( i \), column \( j \) of \( A \) moves to row \( j \), column \( i \) of \( A^T \).

Think of it as reflecting the matrix across its main diagonal: the entry at position \( (1,3) \) swaps with the entry at position \( (3,1) \).

Transpose in Action

A 1 2 3 4 5 6 7 8 9 3 × 3 Transpose Aᵀ 1 4 7 2 5 8 3 6 9 3 × 3

Why Does the Transpose Matter?

Solving Systems
Transposes appear in the normal equations \( A^T A \mathbf{x} = A^T \mathbf{b} \) for least-squares solutions.
Dot Products
The dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) equals the matrix product \( \mathbf{u}^T \mathbf{v} \).
Orthogonality
Orthogonal matrices satisfy \( Q^T Q = I \), making the transpose act as the inverse.

The transpose is one of the most fundamental matrix operations. It connects row space to column space, powers least-squares regression, and is essential to understanding symmetric matrices.

Core Properties of \( A^T \)

1
Double transpose: \( (A^T)^T = A \). Transposing twice returns the original matrix.
2
Sum rule: \( (A + B)^T = A^T + B^T \). Transpose distributes over addition.
3
Scalar rule: \( (cA)^T = cA^T \). Constants pass through the transpose.
4
Product rule: \( (AB)^T = B^T A^T \). The order reverses when transposing a product.

Product Rule: \( (AB)^T = B^T A^T \)

(AB)ᵀ = Bᵀ · Aᵀ Order reverses! first → last last → first

This "reversal" rule extends to any number of factors: \( (ABC)^T = C^T B^T A^T \). Think of it like reversing the order of putting on socks and shoes: to undo, you reverse.

What Is a Symmetric Matrix?

A square matrix \( A \) is symmetric if it equals its own transpose:

Symmetric Matrix 2 7 3 7 5 1 3 1 4 mirror line

Entries mirror across the main diagonal. The 7 at position \( (1,2) \) equals the 7 at position \( (2,1) \).

Properties of Symmetric Matrices

1
Must be square: Only \( n \times n \) matrices can be symmetric, because \( A^T \) must have the same dimensions as \( A \).
2
Real eigenvalues: Every eigenvalue of a real symmetric matrix is a real number.
3
Orthogonal eigenvectors: Eigenvectors from distinct eigenvalues are automatically perpendicular.
4
Spectral theorem: Every symmetric matrix can be diagonalized as \( A = Q \Lambda Q^T \) with \( Q \) orthogonal.

Transpose vs Symmetric

Transpose \( A^T \) Symmetric Matrix
What is it? An operation on any matrix A property of a square matrix
Size requirement Any \( m \times n \) matrix Must be \( n \times n \)
Result size \( n \times m \) (dimensions flip) \( n \times n \) (unchanged)
Key condition \( (A^T)_{ij} = A_{ji} \) \( A = A^T \)
Example Any matrix has a transpose Distance and covariance matrices

Building Symmetric Matrices

Two powerful ways to create a symmetric matrix from any matrix \( A \):

\( A^T A \)
For any matrix \( A \), the product \( A^T A \) is always symmetric. Proof: \( (A^T A)^T = A^T (A^T)^T = A^T A \). ✓
\( A + A^T \)
The sum \( A + A^T \) is always symmetric for any square matrix. Proof: \( (A + A^T)^T = A^T + A = A + A^T \). ✓

What Is Skew-Symmetric?

A square matrix \( A \) is skew-symmetric (or antisymmetric) if \( A^T = -A \). This means \( a_{ij} = -a_{ji} \) for all entries.

Skew-Symmetric 0 3 -5 -3 0 7 5 -7 0

How to Compute \( A^T \)

1
Write the dimensions: If \( A \) is \( m \times n \), then \( A^T \) will be \( n \times m \).
2
Read each row of \( A \): Row 1 of \( A \) becomes Column 1 of \( A^T \), Row 2 becomes Column 2, and so on.
3
Fill in the new matrix: Place each entry \( a_{ij} \) into position \( (j, i) \) of \( A^T \).
4
Verify: Check that diagonal entries stayed in place and off-diagonal entries swapped correctly.

Example 1: Rectangular Transpose

A (2×3) 1 4 7 2 5 8 Aᵀ (3×2) 1 2 4 5 7 8

Example 2: Verifying Symmetry

Is the following matrix symmetric?

B = 4 -1 6 -1 9 2 6 2 3
1
Check: \( b_{12} = \) \(-1\) and \( b_{21} = \) \(-1\). ✓ They match.
2
Check: \( b_{13} = \) \(6\) and \( b_{31} = \) \(6\). ✓ They match.
3
Check: \( b_{23} = \) \(2\) and \( b_{32} = \) \(2\). ✓ All pairs match, so \( B \) is symmetric!

Example 3: Showing \( A^T A \) Is Symmetric

Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \). Compute \( A^T A \) and verify symmetry.

1
Find \( A^T \): \( A^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} \) (a \( 2 \times 3 \) matrix).
2
Multiply \( A^T A \): \( \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} = \begin{bmatrix} 35 & 44 \\ 44 & 56 \end{bmatrix} \)
3
Check symmetry: Entry \( (1,2) = \) 44 and entry \( (2,1) = \) 44. ✓ Symmetric!

Common Mistakes to Avoid

✕ Forgetting to Reverse
Writing \( (AB)^T = A^T B^T \) instead of the correct \( (AB)^T = B^T A^T \). The order must flip.
✕ Non-Square Symmetry
Claiming a \( 2 \times 3 \) matrix is symmetric. Only square matrices can be symmetric since \( A^T \) must have the same size as \( A \).
✕ Dimension Confusion
Forgetting that transposing an \( m \times n \) matrix gives an \( n \times m \) result. The rows and columns swap count.

Quick Reference Summary

Property / Rule Formula
Definition of transpose \( (A^T)_{ij} = A_{ji} \)
Double transpose \( (A^T)^T = A \)
Sum rule \( (A + B)^T = A^T + B^T \)
Scalar rule \( (cA)^T = cA^T \)
Product rule \( (AB)^T = B^T A^T \)
Symmetric definition \( A^T = A \)
Skew-symmetric definition \( A^T = -A \)
Always symmetric \( A^T A \) and \( A + A^T \)

Question 1 of 5

If \( A \) is a \( 3 \times 5 \) matrix, what are the dimensions of \( A^T \)?

Question 2 of 5

Which expression correctly simplifies \( (ABC)^T \)?

Question 3 of 5

A matrix \( A \) is symmetric if and only if:

Question 4 of 5

What must be true about the diagonal entries of a skew-symmetric matrix?

Question 5 of 5

For any matrix \( A \), which of the following is always symmetric?