Reading a connection matrix
A connection matrix has one row and one column for each vertex in a declared order. Each entry records direct connections from the row vertex to the column vertex, including loops and repeated connections
\[M_{ij}=\text{number of direct connections from row }i\text{ to column }j\]
- Fix the vertex order Use the declared order for both rows and columns before reading any connection \(\displaystyle (A,B,C)\text{ gives rows and columns }1,2,3\)
- Read row to column An off-diagonal entry in row i and column j records a direct connection from vertex i to vertex j. \(\displaystyle M_{23}=2\Rightarrow 2\text{ to }3\text{ has two direct connections}\)
- Read the diagonal as loops A leading-diagonal entry records a connection from a vertex to itself only when a loop is present \(\displaystyle M_{22}=1\Rightarrow\text{one loop at vertex }2\)
- Check mirrored entries Equal mirrored entries give equal counts in the two directions. Unequal entries give different counts; check every mirrored pair before calling the whole matrix symmetric \(\displaystyle M_{12}=M_{21}=1\Rightarrow\text{one connection in each direction}\)
Worked example
The vertex order is A, B, C. Use the connection matrix below to identify the connections involving B and count the direct connections arriving at B, counting the loop once
\[M=\begin{bmatrix}0&1&0\\1&1&2\\0&2&0\end{bmatrix}\]
- \(\displaystyle M_{22}=1\)The entry on the diagonal shows one loop at B.
- \(\displaystyle M_{12}=M_{21}=1,\quad M_{23}=M_{32}=2\)The equal mirrored entries show one connection between A and B and two connections between B and C, each in both directions
- \(\displaystyle M_{12}+M_{22}+M_{32}=1+1+2=4\)Adding column 2 counts the connection at B, its loop and the two connections from C.
- \(\displaystyle \text{B has }4\text{ direct connections}\)The total includes the loop once because its matrix entry is 1.
Common mistakes
Treating the diagonal as automatic connections
✕\(\displaystyle M_{ii}=1\text{ for every vertex}\)✓\(\displaystyle M_{ii}=1\text{ only when a loop at vertex }i\text{ is present}\)
A diagonal entry is non-zero only when the network includes a loop at that vertexRecording repeated connections as one
✕\(\displaystyle M_{23}=1\text{ when two direct connections join vertices 2 and 3}\)✓\(\displaystyle M_{23}=2\text{ when two direct connections join vertices 2 and 3}\)
The entry records the number of direct connections, not only whether at least one connection existsReversing the row and column
✕\(\displaystyle M_{23}=1\Rightarrow 3\to2\)✓\(\displaystyle M_{23}=1\Rightarrow 2\to3\)
The row names the starting vertex and the column names the ending vertexDifficulty levels
introMatch a small undirected network with its connection matrix
standardRead one-way links, repeated connections and vertex counts from a four-vertex matrix
challengeInterpret a directed five-vertex matrix containing a loop and repeated connection
examSelect the network whose direct connections match every matrix entry
Questions about this topic
What does a diagonal entry mean in a connection matrix?+
It records a loop from a vertex to itself. It is zero when no loop is present; a vertex reaching itself indirectly is not recorded on the diagonal of the direct connection matrix
How are repeated direct connections recorded?+
The entry is the number of direct connections between the row vertex and the column vertex. For example, two direct connections are recorded as 2.
How can a matrix show a two-way connection?+
A two-way connection contributes equally to the two mirrored entries. Equal non-zero entries show equal counts in the two directions; they can also arise from separately stated one-way links in both directions
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals