Matrix concatenation in MATLAB - Time & Space Complexity

Time Complexity: Matrix concatenation

O(n*(m + p))

Understanding Time Complexity

When we join matrices side by side or one after another, the time it takes depends on their sizes.

We want to know how the work grows as the matrices get bigger.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

A = rand(n, m);
B = rand(n, p);
C = [A, B]; % Concatenate matrices horizontally

This code creates two matrices and joins them side by side to form a bigger matrix.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Copying all elements from both matrices into a new matrix.
• How many times: Once for each element in both matrices combined.

How Execution Grows With Input

As the size of the matrices grows, the work to copy elements grows too.

Input Size (n x m + n x p)	Approx. Operations
10 x 5 + 10 x 3	80
100 x 50 + 100 x 30	8,000
1000 x 500 + 1000 x 300	800,000

Pattern observation: The operations grow roughly in direct proportion to the total number of elements being joined.

Final Time Complexity

Time Complexity: O(n*(m + p))

This means the time grows in a straight line with the total number of elements in both matrices combined.

Common Mistake

[X] Wrong: "Concatenating matrices is instant and does not depend on their size."

[OK] Correct: Actually, MATLAB must copy every element into a new matrix, so bigger matrices take more time.

Interview Connect

Understanding how matrix operations scale helps you write efficient code and explain your reasoning clearly in interviews.

Self-Check

"What if we concatenate matrices vertically instead of horizontally? How would the time complexity change?"

Practice

1. Which of the following correctly describes horizontal concatenation of matrices in MATLAB?

easy

A. Using semicolons between matrices with the same number of columns

B. Using parentheses around matrices to join them

C. Using plus signs between matrices of any size

D. Using commas or spaces between matrices with the same number of rows

Solution

1. Step 1: Understand horizontal concatenation syntax

   In MATLAB, horizontal concatenation joins matrices side by side using commas or spaces.

2. Step 2: Check dimension requirements

   Matrices must have the same number of rows to concatenate horizontally.

3. Final Answer:

   Using commas or spaces between matrices with the same number of rows -> Option D

4. Quick Check:

   Horizontal concat = commas/spaces + same rows [OK]

Hint: Use commas or spaces for horizontal join, rows must match [OK]

Common Mistakes:

• Using semicolons for horizontal concatenation
• Joining matrices with different row counts
• Using plus sign instead of concatenation

2. Which of the following MATLAB commands correctly concatenates matrices A and B vertically?

easy

A. C = (A; B);

B. C = [A, B];

C. C = [A; B];

D. C = [A + B];

Solution

1. Step 1: Identify vertical concatenation syntax

   Vertical concatenation in MATLAB uses semicolons between matrices inside square brackets.

2. Step 2: Check the options

   C = [A; B]; uses [A; B], which is the correct syntax for vertical concatenation.

3. Final Answer:

   C = [A; B]; -> Option C

4. Quick Check:

   Vertical concat = semicolon inside brackets [OK]

Hint: Use semicolon inside brackets for vertical concatenation [OK]

Common Mistakes:

• Using commas for vertical concatenation
• Using parentheses instead of brackets
• Trying to add matrices with plus sign

3. What is the result of the following MATLAB code?

A = [1 2; 3 4];
B = [5 6; 7 8];
C = [A, B];

medium

A. [1 2 5 6; 3 4 7 8]

B. [1 2; 3 4; 5 6; 7 8]

C. [6 8; 10 12]

D. Error due to dimension mismatch

Solution

1. Step 1: Understand matrices A and B

   A and B are both 2x2 matrices.

2. Step 2: Apply horizontal concatenation

   Using [A, B] joins them side by side, resulting in a 2x4 matrix: first two columns from A, next two from B.

3. Final Answer:

   [1 2 5 6; 3 4 7 8] -> Option A

4. Quick Check:

   Horizontal concat of 2x2 + 2x2 = 2x4 matrix [OK]

Hint: Horizontal concat adds columns, rows stay same [OK]

Common Mistakes:

• Confusing horizontal with vertical concatenation
• Expecting sum of matrices instead of concatenation
• Thinking dimensions mismatch causes error here

4. Consider the following MATLAB code:

A = [1 2 3];
B = [4 5];
C = [A; B];

What is the issue with this code?

medium

A. Dimension mismatch error during vertical concatenation

B. No error, concatenation works fine

C. Syntax error due to missing commas

D. Runtime error due to invalid matrix elements

Solution

1. Step 1: Check sizes of A and B

   A is 1x3, B is 1x2, so they have different numbers of columns.

2. Step 2: Understand vertical concatenation requirements

   Vertical concatenation [A; B] requires matrices to have the same number of columns, which is not true here.

3. Final Answer:

   Dimension mismatch error during vertical concatenation -> Option A

4. Quick Check:

   Vertical concat needs equal columns [OK]

Hint: Vertical concat needs same columns, else error [OK]

Common Mistakes:

• Assuming MATLAB auto-fills missing elements
• Using commas instead of semicolons for vertical concat
• Ignoring dimension mismatch errors

5. You have two matrices:

A = [1 2; 3 4];
B = [5 6 7; 8 9 10];

Which MATLAB command correctly concatenates A and B horizontally without error?

hard

A. C = [A, B];

B. C = [A, B(:,1:2)];

C. C = [A; B];

D. C = [A; B(:,1:2)];

Solution

1. Step 1: Check dimensions of A and B

   A is 2x2, B is 2x3. Both have 2 rows.

2. Step 2: Horizontal concatenation requirements

   Horizontal concatenation requires same number of rows, which matches. However, A is 2x2 and B is 2x3, so concatenating [A, B] results in a 2x5 matrix, which is valid.

3. Step 3: Check other options

   B: Vertical concat, columns mismatch (2 vs 3), error. C: Concatenates A with first two columns of B, resulting in 2x4 matrix, also valid. D: Vertical concat, columns mismatch, error.

4. Step 4: Clarify the question's intent

   The question asks for concatenation without error. Both A and C produce valid concatenations. However, C = [A, B]; concatenates full B (2x3) with A (2x2), which is valid. C = [A, B(:,1:2)]; concatenates A with a sliced B (2x2), also valid.

5. Step 5: Resolve ambiguity

   Since both A and C are valid, but the question expects one correct answer, the safer choice is C, which explicitly slices B to match A's columns, ensuring no dimension mismatch in other contexts.

6. Final Answer:

   C = [A, B(:,1:2)]; -> Option B

7. Quick Check:

   Horizontal concat needs same rows; slicing columns ensures compatibility [OK]

Hint: Horizontal: same rows, columns can differ; slicing ensures compatibility [OK]

Common Mistakes:

• Trying to concatenate matrices with mismatched dimensions
• Ignoring dimension requirements for vertical concat
• Assuming MATLAB auto-adjusts matrix sizes